It was shown in

Browder, William. *Fiberings of spheres and $H$-spaces which are rational homology spheres.* Bull. Amer. Math. Soc. $\bf 68$ 1962 202–203. 

that for any fibration $F \to S^n \to B$, where $B$ is a non-trivial polyhedron, $F$ must have the homotopy type of $S^1,S^3$ or $S^7$. This builds on previous work of Spanier-Whitehead and Borel.