We study branched covering spaces in several contexts, proving that under suitable circumstances the cover satisfies the same upper curvature bounds as the base space. The first context is of a branched cover of an arbitrary metric space that satisfies Alexandrov's curvature condition CAT(k), over an arbitrary complete convex subset. The second context is of a certain sort of branched cover of a Riemannian manifold over a family of mutually orthogonal submanifolds. In neither setting do we require that the branching be locally finite. We apply our results to hyperplane complements in several complex manifolds of nonpositive sectional curvature. This implies that two moduli spaces arising in algebraic geometry are aspherical, namely that of the smooth cubic surfaces in complex projective 3-space and that of the smooth complex Enriques surfaces.