The Davis—Kahan—Wedin theorem tells us that a matrix’s singular vectors are unstable under perturbation when the corresponding singular values are close together. This fact has major implications for many randomized matrix algorithms, many of which rely on a random sketch of a large matrix to approximately recover that matrix’s dominant singular subspaces. If there isn’t a large gap between the leading and trailing singular values of the matrix being sketched, then existing theory, both deterministic and randomized, tells us that the singular subspaces of the sketch can’t be trusted to provide an accurate approximation of the true singular subspaces. We present a new, exact quantification of the angular error from randomized subspace approximation via sketching, which tells us when we can and cannot expect good subspace approximations. One surprising consequence is that for matrices whose leading and trailing singular values have fixed Frobenius norm, a larger singular value gap produces a worse subspace approximation.