Let $X,Y$ be separable and metrizable, with $Y$ Polish, and suppose there is a topological quotient map $f:X\to Y$ with compact fibers. Is $X$ Polish?
I have a specific space in mind, so if the answer is no and some extra criterion comes to mind ($Y$ is not locally compact, and $f$ is not a fiber bundle), I would appreciate it.
I think the answer is no. Consider the following subsets of the Euclidean plane.
Let $X_0 = \{(p,q)\in (\mathbb R\setminus \mathbb Q)\times \mathbb R: 0\le p \le 1, 0 \le q \le 1 \}$, $X_1 = \{(p,q)\in ( \mathbb Q)\times \mathbb R: 0\le p \le 1, -1 \le q \le 0 \}$,
Let $X=X_0\cup X_1$, $Y=[0,1]$, and let $f$ be the projection.
(ADDED: $X$ is not Polish, as $X$ contains a closed subset homeomorphic to $\mathbb Q$. Closed subsets of Polish spaces are Polish, but $\mathbb Q$ is not.)
$X$ need not be Polish; here's a counterexample:
Let $Y$ be the real line with its usual topology, and let $A$ be an arbitrary subset of $Y$. (I'll specialize $A$ later, but for now, let it be arbitrary.) Let $$ X=\{(a,b)\in Y\times[0,1]:b=0\text{ or }a\in A\}. $$ Then the projection $(a,b)\mapsto a$ is a topological quotient map. Its fibers are copies of $[0,1]$ (over the points in $A$) and singletons (over the other points of $Y$), so they are compact. $X$ is a separable metric space, a subspace of $Y\times[0,1]$.
What remains is to see whether $X$ is Polish. Since it's a subspace of $Y\times[0,1]$, it will be metrizable by a complete metric (and therefore Polish) if and only if it is a $G_\delta$ subset of $Y\times[0,1]$. There are only $\mathfrak c$ (the cardinality of the continuum $G_\delta$ subsets of $Y\times[0,1]$, and there are $2^{\mathfrak c}$ different possibilities for $A$, all yielding different sets $X$. So, for most choices of $A$, the resulting $X$ sill not be Polish.
