Let $M$ be a compact closed path-connected manifold.

There is the Postnikov tower as in

http://en.wikipedia.org/wiki/Postnikov_system

which tells us $M$ can be realized as an inverse limit of a tower of fibrations, up to homotopy, and this tower is often infinite, if $\pi_n(M)$ does not vanish when $n$ is large enough.

So I wonder, what if we work in the homeomorphic category instead of the homotopy category, and try to realize $M$ as a fiber bundle

$F \rightarrow M \rightarrow B$

instead of a fibration (which is more general). Are there certain obstructions to this, such that one can tell whether this is possible by looking at the algebraic topology of $M$?

As an example of what I have in mind, note we have Hopf fibrations for $S^3, S^7, S^{15}$, and these are the only possibilities if the fiber $F$ and the base space $B$ are required to be spheres as well. I wonder what other spheres can be realized as non-trivial fiber bundles, if we don't put any restrictions on $F$ and $B$.

Thank you very much.