Here are two classical results which depend on whether a parameter is 2 or 3:

It is possible to bisect an arbitrary angle with ruler and compass, but impossible to trisect it.

While there are infinitely many Pythagorean triples, i.e. integer solutions to $x^2+y^2=z^2$, there are no non-trivial integer solutions to $x^3+y^3=z^3$.

There are several other instances where the dividing line seems to be between 2 and 3:

A 2-regular tree is countable, a 3-regular tree is uncountable.

2SAT is solvable in polynomial time, 3SAT is NP-complete.

A random walk on $\mathbf Z^2$ is recurrent, while a random walk on $\mathbf Z^3$ is transient.

What other examples can you think of?