Topologists are often very good at explaining the geometric intuition behind certain results and programs of research. For instance, the particular interest in 4 manifolds is often explained by saying "4 dimensions are enough to allow for certain constructions but not enough to allow one to undo those constructions" (or something like that).
What I'm wondering is, does there exist any sort of intuition for why certain topological manifolds do not admit smooth structures or triangulations. This fact is extremely contrary to my naive intuition about manifolds, so I was hoping there might exist some counter-intuition behind it.