Has a Serre fibration $f:E\to B$ with $B$ a connected space isomorphic fibers over different points of $B$?
If $f$ is a fiber bundle, then all fibers are isomorphic. Hence, a possible counterexample would be a Serre fibration which is not a fiber bundle. It would be nice, if someone could provide a counterexample where $E$ and $B$ are non-pathological spaces, e.g. CW complexes.