We study asymptotic inference based on cluster-robust variance estimators for regression models with clustered errors, focusing on the wild cluster bootstrap and the ordinary wild bootstrap. We state conditions under which both asymptotic and bootstrap tests and confidence intervals will be asymptotically valid. These conditions put limits on the rates at which the cluster sizes can increase as the number of clusters tends to infinity. To include power in the analysis, we allow the data to be generated under sequences of local alternatives. Under a somewhat stronger set of conditions, we also derive formal Edgeworth expansions for the asymptotic and bootstrap test statistics. Simulation experiments illustrate the theoretical results, and the Edgeworth expansions explain the overrejection of the asymptotic test and shed light on the choice of auxiliary distribution for the wild bootstrap.