In a sense you more or less know what non-compact manifolds must look like provided you have a classification of the compact manifolds. The starting point is the basic observation (Whitney) that a non-compact manifold has a proper function f : M --> R. So the preimage of the intervals [-n,n] for n=1,2,3,... form a nested family of compact submanifolds of M that exhaust the manifold -- provided f is transverse to the integers, which can be accomplished.
So understanding M boils down to seeing how the features of this family of submanifolds "pile up", and putting some sort of reasonable ideal boundary on M -- since "ideal boundary" is a lower-dimensional phenomenon, in principle you might be inclined to think this is reasonable.
Of course I'm being pretty vague but it sounds like you were looking for something like this?
On the other side of things, because of the above non-compact manifolds are decidedly less combinatorial objects. You don't have finite, combinatorial descriptions. Larry Siebenmann shows people the example of the smooth structure on IxR^4 such that the projection map IxR^4-->I is a smooth submersion, but for which the fibers a pairwise non-diffeomorphic exotic R^4's.