A short paper with references to several early counterexamples proves that (in the good category of compactly generated weak Hausdorff spaces) a Serre fibration in which the total space and base space are both CW complexes is necessarily a Hurewicz fibration.  

* M. Steinberger and J. West. _Covering homotopy properties of maps between CW complexes or ANRs_. Proc. Amer. Math. Soc. **92** (1984), 573–577, doi:[10.1090/S0002-9939-1984-0760948-6](https://doi.org/10.1090/S0002-9939-1984-0760948-6).  

(The proof is corrected in R. Cauty. _Sur les overts des CW-complexes et les fibr\'es de Serre_.  Colloquy Math. **63** (1992), 1–7, [link](https://bibliotekanauki.pl/articles/967590)).  No relationship between the covering map and the CW structures is required.  This argues either that counterexamples are 
pathological or that it is a special property for the total space of a Serre fibration with CW base space to be a CW complex, although it has the homotopy type of a CW complex if the fiber does.