I came across the concept of fiber bundles as manifolds having locally the structure of a product. My question is the following. Suppose you are given a manifold $M$ but does not know if it has a "product structure". Are there ways, given a manifold M, to somehow "reverse"engineer if the manifold has a product structure, that is, if and when it is possible for $M$ to be homeomorphic to the total space $E$ of a bundle, $\pi : E \rightarrow N$ (of course $N$ must be found, along the projection $\pi$).
Does the question make some sense?
Random thoughts: in a fiber bundle $\pi : E \rightarrow N$ a local trivialization provides a homeomorphism $\pi^{-1}(U)\rightarrow U \times R^k$. From here we have probably local coordinates of the form $(x_1,\dots,x_n,v_1,\ldots,v_k)$, the first $n$ coordinates coming from a local chart in $M$ and the last $k$ from the fiber.
Instead, given a manifold $M$ a local chart would provide some coordinates $(y_1,\ldots,y_r)$. Can we somehow identify the $y_1,\ldots,y_t$ with $t<r$ as coordinates of a smaller manifold?