I think none of the above posts answer the question "is every differentiable manifold diffeomorphic to an open submanifold of a compact one?". Rather they answer "is every differentiable manifold diffeomorphic to the interior of a compact one?" The reason for the confusion could be the latter question is fundamental in geometric topology, while the former one has little significance. Anyway,

The connected sum $V$ of infinitely many copies of $CP^3$'s is not diffeomorphic to an open subset of a compact manifold.

*EDIT: Hats off to Torsten Ekedahl who pointed out in comments that my argument below is incorrect (thus I don't know whether the above statement about $V$ is true). I decided not to delete it because it illuminates some subtleties of the original question.*

The point is that any diffeomorphism onto an open subset pulls back the tangent bundle, and in particular, pulls back the first Pontryagin class $p_1$. Thus if $V$ is an open subset of a compact manifold $M$, then its first Pontryagin class $p_1(V)$ lies in the image of $H^4(M)\to H^4(V)$, which is a **finitely generated** subgroup of $H^4(V)$, which is the infinite product of $\mathbb Z$'s corresponding to generators of $H^4(CP^3)$. The first Pontryagin class of $CP^3$ is a multiple of a generator of $H^4(CP^3)\cong\mathbb Z$, and removing a finite set of points from $CP^3$ does not affect the $4$th skeleton, so $p_1(V)$ does not lie in a finitely generated subgroup of $H^4(V)$.

I am curious to see low-dimensional answers to the question "is every differentiable manifold diffeomorphic to an open submanifold of a compact one?"

openembedding into a compact one. I presume it's easy to embed any manifold as a locally closed submanifold of a sphere: just Whitney-embed as a closed submanifold of $\mathbb{R}^N$, send $\mathbb{R}^N$ diffeomerphically onto a ball, which sits inside a larger ball, and one-point-compactify the larger ball. $\endgroup$ – Qfwfq Apr 25 '10 at 4:16