We know that covering spaces of many low dimensional CW complexes such as graph (CW structures with only 0-cells and 1-cells) and all compact surfaces has CW structure. [The second fact is due to Uniformation Theorem from theory of Riemann surfaces. Basically, the only simply connected 2-manifold are the sphere, the Euclidean plane and hyperbolic plane.]
So I wonder, as put forth in the title, if this is true in general. That is, if $X$ is connected, locally path-connected and it has a CW complex structure then does its universal covering space also have a CW complex structure?