Barratt and Milnor (_An Example of Anomalous Singular Homology_) proved that (for $n > 1$) the singular homology of the union of countably many $n$-spheres with one point in common and radii tending to $0$ is non-trivial in arbitrarily high dimensions. Thus any CW-replacement of this space is infinite-dimensional. On the other hand, it is a closed subspace of $\mathbb{R}^{n+1}$ so its covering dimension is at most $n+1$.