The sphere-packing problem, already mentioned in other answers, has some bizarre behavior as the number of dimensions changes. Results in the Leech lattice lead to "monstrous moonshine", and the bizarreness of the sporadic groups. Meanwhile, in string theory, its known that certain things can only happen in 26 or 10 dimensions. Hopf fibrations show you how to deconstruct 3-D space into 2+1 dimensional space, or 11 dimensions into 7+4. But why can't this work in general N, you might wonder? The "homotopy groups of spheres" indicate that even the common-sense notion of a sphere changes from dimension to dimension.