I'm interested in examples of situations that are easier in higher dimensions. To give a flavor of what I am looking for, here are two of my favorites:
(a) In dimensions 2 and higher, one can characterize the standard normal distribution (up to a constant multiple) as the spherically symmetric distribution with independent marginals. Obviously, such a characterization fails miserably in one dimension.
(b) In dimension 3 and higher, you can prove Desargues' Theorem using the incidence axioms. No such proof works in two dimensions (and there are non-Desarguesian planes).
What are some other nice results where having at least $n$ dimensions allows one to prove things or characterize things in ways that are not possible in fewer than $n$ dimensions?