The following question was something that came to my mind during my (unsuccessful) attempt at answering this MO-question.

Let $X$ be a topological space, and let $\tilde{X}\to X$ be a CW-approximation. Given that $X$ has covering dimension $n$, can anything be said about the covering dimension of the CW-approximation $\tilde{X}$? If it's not generally true that $\dim\tilde{X}\leq\dim X$, is it possible to give explicit examples?