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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. (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). 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.