This question was studied in a <a href="http://www.jstor.org/stable/1993612">paper</a> by Browder who proves the following. 

<b>Thereom.</b> Consider any Serre fibration of $F\to S^n\to B$ where $F$, $B$ are connected polyhedra. Then $F$ is homotopy equivalent to $S^1$, $S^3$, or $S^7$. If $F=S^1$, then $B$ is homotopy equivalent to $CP^k$ with $2k+1=n$. If $F=S^7$, then $B$ is homotopy equivalent to $S^8$. 

One expects that if $F= S^3$, then $B$ is homotopic to a quaternionic projective space of a suitable dimension but Browder mentions (if I am reading is correctly) that this is not true and there are other examples. 

<i> Caution:</i> there must be something wrong with the above theorem because it rules out existence of $S^7$ bundles over octonian projective plane. Can someone explain the situation? I care because Browder's theorem was used in some geometric problems involving boundary at infinity of nonnegatively curved manifolds.