It is well-known and easy to prove (see for instance this post) that every smooth manifold admits a "good cover", i.e. a locally finite cover by open balls such that all nonempty intersections of the elements of this cover are still homeomorphic to open balls. The same is true, more generally, for PL manifolds.

Question: Do all topological $n$-manifolds admit good covers?

I am especially curious about $n=4$.

Here is my guess: a good cover is too close to a handle decomposition for gauge-theoretic obstructions in dimension 4. In higher dimensions- I do not even have a guess.