In this paper we are interested in inference based on heteroskedasticity consistent covariance matrix estimators, for which the appropriate bootstrap is a version of the wild bootstrap. Simulation results, obtained by a new very efficient method, show that all wild bootstrap tests exhibit substantial size distortion if the error terms are skewed and strongly heteroskedastic. The distortion is however less, sometimes much less, if one uses a version of the wild bootstrap, belonging to a class we call ``tamed'', which benefit from an asymptotic refinement related to the asymptotic independence of the bootstrapped test statistic and the bootstrap DGP. This version always gives better results than the version usually recommended in the literature, and gives exact results for some specific cases. However, when exact results are not available, we find that the rate of convergence to zero of the size distortion of wild bootstrap tests is not very rapid: in some cases, significant size distortion still remains for samples of size~100.
