Let
$$
F\to E\xrightarrow{\pi} B
$$
be a fibre bundle over a connected and compact base $B$.

Is it true that the total space $E$ has the homotopy type of a CW complex, if the fibre $F$ and the base $B$ have the homotopy type of a CW complex?

I don't think that this is true for Serre fibrations $\pi$, but I don't know a proof either.