Especially if you're looking at smooth manifolds, there's some weird stuff that happens in the noncompact case. The most famous example is the existence of infinitely many [manifolds][1] that are homeomorphic but not diffeomorphic to R^4. 

I don't know that similarly weird things happen just for topological manifolds, but I wouldn't count it out.

I guess it's worth mentioning, though, that plenty of utterly crazy things happen even for compact topological 4-manifolds. For instance, you have the E8 manifold, which isn't triangulable, and on the other end you have manifolds that admit way too many piecewise linear structures. This weirdness disappears for compact manifolds of higher dimension (fortunately), but for non-compact manifold you have to deal with stuff like R x E8, which I suspect isn't much nicer.


  [1]: http://en.wikipedia.org/wiki/Exotic_R4