Results true in a dimension and false for higher dimensions Some theorems are true in vector spaces or in manifolds for a given dimension $n$ but become false in higher dimensions.
Here are two examples:


*

*A positive polynomial not reaching its infimum. Impossible in dimension $1$ and possible in dimension $2$ or more. See more details here.

*A compact convex set whose set of extreme points is not closed. Impossible in dimension $2$ and possible in dimension $3$ or more. See more details here.


What are other "interesting" results falling in the same category?
 A: Von Neumann's inequality, that $\|p(A)\| \leq \|p\|_\infty$ for any polynomial $p$ and Hilbert-space contraction $A$ ($\|p\|_\infty$ being the maximum modulus of $p$ on the unit disk) holds in dimensions one (von Neumann) and two (Ando), but not for three and higher (Varopoulos).
A: Due to Kuratowski:

For any partition $\mathbb{R}^n=\bigcup_{i=1}^nA_i$ there exists an $i \in \{1,\dots,n\}$ and a straight line $l$ parallel to the $i$-th axis
  such that $A_i\cap l$ is infinite.

This is true for $n \leq m$ and false for $n>m$ where $|\mathbb{R}|=\aleph_{m-1}$.
A: Compact manifolds in dimension $2$ and $3$ which carry a metric of negative sectional curvature also carry an hyperbolic structure. 
False in dimensions $\geq 4$, where counterexamples have been found by Mostow and Siu, and later by Gromov and Thurston. 
A: True for dimensions $n\le 8$: the existence of left-invariant affine structures on nilpotent Lie groups of dimension $n$. All nilpotent Lie groups of dimension $n\le 8$ admit such a structure. This is probably false for all $n\ge 9$, but this is only proved for $n=10,11,12$ so far. More precisely, there exist nilpotent Lie groups of dimension $10,11,12$ which do not admit any affine structure.
A: True only in dimension 2: the 2-sphere with any smooth Riemannian metric with positive sectional curvature can be isometrically realized as convex hypersurface in the Euclidean space $\mathbb{R}^3$. For higher dimensional spheres analogous statement is not true.
A: Keller's conjecture asserts that whenever one tiles ${\bf R}^n$ by unit cubes, there must be two cubes which share a common face.  True when $n \leq 6$, false for $n\geq 8$, and still open for $n=7$.
UPDATE, October 2019: It appears that the conjecture has now been resolved in the affirmative by computer assisted proof in $n=7$: https://arxiv.org/abs/1910.03740
A: The Schoenflies theorem: if $\Sigma^n \subset \Bbb R^{n+1}$ is homeomorphic to $S^n$, then there is a homeomorphism $h \colon \Bbb R^{n+1} \to \Bbb R^{n+1}$ such that $h(\Sigma^n) = S^n$ (the round $n$-sphere). Trivial for $n = 0$, true (but hard) for $n = 1$, false for $n \geq 2$ (counterexample: the Alexander horned sphere).
A: An n-dimensional brownian motion visits every neighborhood of $\mathbb{R}^n$ infinitely often with probability 1 iff $n \leq 2$
A: $\mathbb R^n$ endowed with the $1$-norm has the binary intersection property if and only if $n=1$ or $n=2$, cf here for details.
A: For all $n > 3$, there is no non-trivial $1$-dim. knot included in $\mathbb{R}^n$.
Edit (sept. 22, 2014)
More generally, $\forall  n > r+2$, there is no non-trivial piecewise-linear $r$-dim. knot included in $\mathbb{R}^n$,
and $\forall n > (3r+3)/2$, there is no non-trivial smooth $r$-dim. knot included in $\mathbb{R}^n$ (see this wiki page).  
A: The Theorem by Poincare and Benedixson: 
Given an autonomous differential equation $\dot{x}=f(x)$ on some $U\subset\mathbb{R}^2$ with initial conditions, such that the solution $u(x)$ exists for all $t>0$, then any compact $\omega$-limit set with finitely many critical points is one of the following:


*

*A Critical Point

*A periodic orbit

*a connected set composed of a finite number of fixed points together with homoclinic and heteroclinic orbits connecting these.


Polemically simplified: 2-dimensional autonomous systems are not too chaotic. 
The Theorem does not hold in 3 or more dimensions because the Jordan curve theorem does not hold there.
A: Claim: Every smooth $n$-dimensional manifold homeomorphic to a sphere is also diffeomorphic to a sphere.  In other words, there are no exotic spheres of dimension $n$.
True for $n=1,2,3,5,6,12,61$.  Open for $n=4$.  False for all other $n < 126$, and for all odd $n \geq 126$ (according to forthcoming work of Behren, Hill, Hopkins, and Mahowald, plus earlier results; there are some additional results for large even $n$ also, but I don't know the precise statements).  The $n=7$ case was a famous counterexample of Milnor.  The problem is closely connected to that of determining the (stable) homotopy groups of spheres, see e.g. the previous MathOverflow question Exotic spheres and stable homotopy in all large dimensions? .
A: My favorite example (by some distance) is the problem in discrete geometry often called Borsuk's conjecture. The basic question dates back to the 1930s and could be explained to a child: is every bounded subset of $\mathbb{R}^d$ decomposable into $d+1$ subsets of strictly smaller diameter? 
You could draw figures all day and convince yourself that the answer is yes in dimensions $2$ and maybe even $3$. But then you would read the following paper (a good candidate for maximizing the ratio of importance to length):

Jeff Kahn and Gil Kalai, A counterexample to Borsuk's conjecture, Bulletin of the American Mathematical Society 29 (1993), 60–62.

As far as I can recall, the best lower bound on $d$ for which the conjecture is known to be false right now is $64$ or $65$.

Update (9/19/14): The current bound is $64$, if a preprint from January of this year is to be believed:

T Jenrich  and AE Brouwer, A 64-dimensional counterexample to Borsuk’s
  conjecture.

The $65$-dimensional example which they use as a starting point is due to A Bondarenko.
A: From graph theory:
Embed a fully-connected graph without multiple edges into a 2-dimensional euclidean space (the plane), some edges will intersect each-other if the number of vertices is at least a small constant (6), however, the edges will not intersect each-other for any number of vertices if the same graph is embedded into an euclidean space of dimension $n \geq 3$.
There is a weaker version of the above statement:
Embed an arbitrary graph without multiple edges with minimum degree of 4 into a 2-dimensional euclidean space, some edges will intersect each-other if the number of vertices is at least a small constant (5), however, the edges will not intersect each-other if the same graph is embedded into an euclidean space of dimension $n \geq 3$.
There are similar results for spherical, cylindrical, toroidal and other  topological spaces.
A: Riemann mapping theorem is true for (complex) dimension $1$, but is false for dimension greater or equal to $2$.
A: Reccurence of the random walk on $\mathbb{Z}^2$ implies the following:  If two random walkers (say two lovers) are walking on a $2$-dimensional grid then they will eventually meet.
In dimension $3$ this isn't true, however the following property (let's call it the perfume property) is true:  Eventually one of the lovers will be at a place that the other one visited before him (and hence be able to smell their perfume).  In fact this will happen infinitely many times.
The perfume property is satisfied by $\mathbb{Z}^d$ if and only if $d = 1,2,3,\text{ or }4$.  On $\mathbb{Z}^5$ there's a positive probability that the lovers will never even catch each other's scent.
A: This is only true in dimension two:
Let $f$ be a harmonic function on a Riemannian manifold $(M,g)$.
If the metric $g$ is changed conformally, the function $f$ is still harmonic.
(The analogous result is true in any dimension $n=\dim M$ if one considers $n$-harmonic functions instead of harmonic ones, but the only case corresponding to the linear 2-Laplace equation is two dimensional.)
A: What about the symmetric group $S_n$ being solvable if and only if $n<5$ (it can be stated as a dimension phenomenon, if one really really wants to)
A: All two-dimensional manifolds are locally conformally flat thanks to the existence of isothermal coordinates.  But this is not true in dimension greater or equal to 3. There exists manifolds which are not locally conformally flat in dimension greater or equal to 3. See here.
A: A set X is said to be m-convex m>=2 if for every m distinct points in X at least one of the line segments determined by those points belongs to X.
For compact sets and m=3 the decomposition into convex, aka 2-convex ,sets has been known for many years. In the plane F.A.Valentine of UCLA showed in 1957 that every 3-convex set was the union of 3 convex sets , see the 5 pointed star on the USA flag. H.G.Eggleston in 1976 gave an example of a compact 3-convex set in  R4 which was not the union of finitely many convex sets. In R3 some years ago I outlined an easy proof in sci.math.research that in R3 a compact 3 convex set was the union of 4 convex sets, using the 4 colour theorem of graph theory.
For information but off topic : Many years ago much was published in the Israel Journal Of Mathematics on planar decomposition, into convex sets ,   bounds  as  a function of m for  compact m-convex sets. There was also a comprehensive treatment by M. Breen of planar non-closed 3-convex sets in 1977. I know of no associated higher dimension work.
A: By  Liouville's theorem, a bounded subharmonic function in $\mathbb{R}^n$ is a constant. This holds not true if $n\ge 3$. 
There are many similar corresponding facts in potential theory.
A: $S^{2}$   and  $\mathbb{R}^{2}$ satisfies the Poincare  Bendixon theorem  but this theorem is not satisfied by higher dimensional spheres  or  Euclidean spaces.
For  a  related MSE post see the following.
https://math.stackexchange.com/questions/861231/extension-of-poincar%C3%A9-bendixson-theorem-to-mathbbr3
As another example: For $n>8$ there is  no a $n$-dimensional subvector  space of $M_{n}(\mathbb{R})$ which all non zero elements are invertible matrix. The only possible $n$ are $n=1,2,4,8$. Such subvector spaces correspond to matrix representation of real numbers, complex numbers, Quaternions  and  Cayley numbers, respectively.
A: The Schrödinger operator
$$
-\Delta + gV(x)
$$
in $L^2(\mathbb R^d)$ with $V\le 0$, $V\not\equiv 0$ (and, let's say, $V\in C_0^{\infty}$) has a negative eigenvalue for any $g>0$: true in $d=1, 2$, false for $d\ge 3$. (In $d\leq 2$, consider the scaling $\phi_\lambda(x) = \phi(x/\lambda)$ with $\lambda \nearrow \infty$. In $d \geq 3$, consider, for example, Hardy's inequality.) 
In more physical terms: in $d=1, 2$, any attractive force, no matter how weak, can bind a particle; this is not true in $d\ge 3$.
A: Here's one where the only exceptional dimension is 11. 
Let $A$ be an $n\times n$ matrix with non-negative entries. 
The smallest positive integer $r$ such that $A^r$ has only positive entries (if there is such an $r$) is denoted $\gamma(A)$. 
Lewin and Vitek conjectured that for every $n$ and every $r<1+(n^2-2n+2)/2$ there is an $n\times n$ matrix $A$ with $\gamma(A)=r$. 
Zhang proved that $n=11$ is the only exception to this conjecture. 
More history, and complete bibliographic references, at https://math.stackexchange.com/questions/450090/if-p-is-a-regular-transition-probability-matrix-then-pn2-has-no-zero-ele
A: The sausage conjecture:
Which way of arranging $M$ unit balls in $\mathbb{R}^n$  minimises the content of their convex hull? For $M$ small the answer is always to arrange them along a line, so that their convex hull is a "sausage". When $n\geq42$ this continues to be true for arbitarily large $M$.
But when $n\leq4$ it eventually becomes better to put the balls in a big round "meatball".
The cases $n=5,\dots,41$ are (as far as I know) open. The conjecture is that in these cases the sausage remains optimal.
A: A series of essentially equivalent statements on zonoids which hold for $d=2$ but fail for $d\geq 3$:
any convex symmetric polytope in $\mathbb{R}^d$ is a Minkowski sum of segments;
any convex symmetric body in $\mathbb{R}^d$ is a section of a unit ball in $L^1$-type space (for polytope finite-dimensional hyperoctahedron  is enough);
any convex symmetric body in $\mathbb{R}^d$ is a projection of a unit ball in $L^{\infty}$-type space (for polytope finite-dimensional cube is enough);
any Banach norm in $\mathbb{R}^d$ may be expressed as $\|x\|=\int |(x,y)| d\mu(y)$ for some Borel measure $\mu$ on $\mathbb{R}^d$, where $(x,y)$ is a scalar product;
for any norm $\|\cdot\|$ on $\mathbb{R}^d$ and any vectors $v_1,u_1,\dots,v_n,u_n$ we have $\sum \|u_i\|\geq \sum \|v_i\|$ provided that $\sum |(u_i,y)|\geq \sum |(v_i,y)|$ for any vector $y$.
A: The James-Stein Estimator.
Suppose $Y \in \mathbb{R}^n$ is a Gaussian vector with unknown mean $\mu$ and known spherical variance $\sigma^2I$. Given an observation of $Y$, we are interested in finding an estimator $\hat{\mu}$ of $\mu$ which minimizes the expected mean squared risk $R(\hat{\mu}) := E(||\hat{\mu} - \mu||^2)$. 
For $n \le 2$, the Gauss-Markov estimator, $\hat{\mu}_{GM} = Y$ minimizes $R(\hat{\mu})$. But for $n \ge 3$ the James-Stein estimator
$$
\hat{\mu}_{JS} = \left(1 - \frac{(n-2)\sigma^2}{||Y||^2}\right)Y
$$
satisfies $R(\hat{\mu}_{JS}) \le R(\hat{\mu}_{GM})$. Note however, that the James-Stein estimator itself does not minimize the mean squared risk.
A: If $F=(f_\lambda)_{\lambda\in\Lambda}$, is an analytic family of quadratic-like maps (with some conditions), and $M_F$ is the set of $f_\lambda$ with connected Julia set, then there is a homeomorphism of $M_F$ with the Mandelbrot set $\{c\in\mathbb{C}:f_c(z)=z^2+c\text{ has connected Julia set}\}$. See http://www.math.cornell.edu/~hubbard/PolyLikeMaps.pdf. Douady called this the "miracle of continuity" coming from the measurable Riemann mapping theorem.
Analytic families of polynomial-like maps of degree $\geq 3$ have discontinuous straighting maps. See http://arxiv.org/pdf/0903.4289v2.pdf.
A: Banach–Tarski paradoxical decomposition of the unit ball.
A: Bernstein's Problem/Theorem: a graphical minimal hypersurface of $\mathbb{R}^n$ is necessarily a hyperplane. True only for $n\leq 8$. 
Related: the De Giorgi conjecture (now known to be true) concerning the geometry of level sets of bounded solutions to a certain nonlinear elliptic PDE. 
A: There is a proper subdomain of $\mathbb{C}^n$ biholomorphically equivalent to $\mathbb{C}^n$ (a Fatou-Bieberbach domain).
False for $n=1$, true for $n \geq 2$.
A: Cross product version of Hurwitz's 1, 2, 4, 8 theorem
There exists a 'cross product' on $\mathbb{R}^n$ if and only if $n=1, 3, 7$.
A cross product means a bilinear map $*: \mathbb{R}^n\times \mathbb{R}^n\rightarrow \mathbb{R}^n$ with the following usual properties:  
(i) The map $(x,y,z)\mapsto(x*y)\cdot z$ is trilinear and alternating.
(ii) $(x*y)*x=(x\cdot x)y-(x\cdot y)x$
Where $\cdot$ denotes the usual dot product on $\mathbb{R}^n$.
For $n=1, 3, 7$ such products are induced by multiplication $x*y:=\frac12(xy-yx)$  where $xy$ is the multiplication of pure elements $x$ and $y$ of complex, quaternion and Cayley algebra over $\mathbb{R}$. 
A: Busemann-Petty problem (see http://en.wikipedia.org/wiki/Busemann%E2%80%93Petty_problem) has positive solution only in dimension at most 4. The problem says: assume that in the Euclidean space $\mathbb{R}^n$ one has two convex centrally symmetric bodies $K$ and $L$. Assume that for any linear hyperplane $H$
$$vol_{n-1}(K\cap H)\leq vol_{n-1}(L\cap H).$$
Is it true that $vol(K)\leq vol(L)$?
The answer is positive for arbitrary convex centrally symmetric bodies $K,L\subset \mathbb{R}^n$ as above if and only if $n\leq 4$.
A: Hilbert's third problem: are two polyhedra (in Euclidean $n$-space) of the same volume scissors congruent? This is true in dimensions $\leq 2$. In dimensions $\geq 3$, an additional invariant of scissors congruence classes of polyhedra is needed: the Dehn invariant (which Dehn used in his solution of Hilbert's third problem). Volume and Dehn invariant fully characterize scissors congruence of polyhedra in dimensions 3 (Sydler) and 4 (Jessen). Unfortunately, it is not known in higher dimensions if these are the only invariants...
As a concrete consequence: in dimension $2$, we can give an elementary definition of area for polygons, based on cutting and pasting to get to rectangular shape. The negative solution to Hilbert's problem implies that such an elementary definition of volume of polyhedra is impossible in higher dimensions.  
See also this MO-question for a related phenomenon where the Dehn invariant constrains tilings of euclidean space in dimensions $\geq 3$.
A: The three altitudes of a triangle intersect in a common point, the orthocenter of the triangle.
This does not generalize to tetrahedra and higher-dimensional simplices. The altitudes of a general $d$-simplex, $d \geq 3$, are not necessarily concurrent.
Source: 
Hajja, Mowaffaq; Martini, Horst. Orthocentric simplices as the true generalizations of triangles. Math. Intelligencer 35 (2013), no. 2, 16--28
A: For every set of $m$ vectors in ${\mathbb R}^d$ such that the angle between every two of them is at most $\pi/2$, there is an orthogonal transformation that maps all vectors to the positive orthant ${\mathbb R}^d_{\geq 0}$.
The statement is true for $d\leq 2$ and false for $d \geq 3$. If we require additionally that $m = d$, the statement is true for $d\leq 4$ and false for $d \geq 5$.  
References


*

*Gray, L. J. and Wilson, D. G. Nonnegative Factorization of Positive Semidefinite Nonnegative Matrices. Linear Algebra Appl. Appl. 31, 119-127, 1980. 

*Xu, Changqing. Completely Positive Matrix. From MathWorld—A Wolfram Web Resource, created by Eric W. Weisstein. http://mathworld.wolfram.com/CompletelyPositiveMatrix.html 
A: It's interesting that no one has so far mentioned Pythagorean triples and Fermat's Last Theorem :)
EDIT: With sufficient imagination any integer can be interpreted as the dimension of an appropriate vector space. However, in this particular case hardly any imagination is necessary: it is in dimensions 1 and 2 only that there exists 3 hypercubes with integer sides such that the sum of volumes of the first two is equal to the volume of the third one.
A: Let $B_2^n$ denote the Euclidean unit ball in $\mathbb{R}^n$. Then the Minkowski-sum $B_2^{n-1}+B_2^n$ is a zonoid whose polar is also a zonoid for $n\leq 4$, but not for $n\geq 6$. (here).
P.S -- I never did get to check the case $n=5$.
A: Theorem(Samelson). The only Euclidean spheres that can be made into topological groups are $\mathbb S^0,\mathbb S^1$ and $\mathbb S^3$. 
