Phylogenetic bootstrapping (BS) is a standard technique for inferring confidence values on phylogenetic trees that is based on reconstructing many trees from minor variations of the input data, trees called replicates. BS is used with all phylogenetic reconstruction approaches, but we focus here on one of the most popular, maximum likelihood (ML). Because ML inference is so computationally demanding, it has proved too expensive to date to assess the impact of the number of replicates used in BS on the relative accuracy of the support values. For the same reason, a rather small number (typically 100) of BS replicates are computed in real-world studies. Stamatakis et al. recently introduced a BS algorithm that is 1 to 2 orders of magnitude faster than previous techniques, while yielding qualitatively comparable support values, making an experimental study possible. In this article, we propose stopping criteria--that is, thresholds computed at runtime to determine when enough replicates have been generated--and we report on the first large-scale experimental study to assess the effect of the number of replicates on the quality of support values, including the performance of our proposed criteria. We run our tests on 17 diverse real-world DNA--single-gene as well as multi-gene--datasets, which include 125-2,554 taxa. We find that our stopping criteria typically stop computations after 100-500 replicates (although the most conservative criterion may continue for several thousand replicates) while producing support values that correlate at better than 99.5% with the reference values on the best ML trees. Significantly, we also find that the stopping criteria can recommend very different numbers of replicates for different datasets of comparable sizes. Our results are thus twofold: (i) they give the first experimental assessment of the effect of the number of BS replicates on the quality of support values returned through BS, and (ii) they validate our proposals for stopping criteria. Practitioners will no longer have to enter a guess nor worry about the quality of support values; moreover, with most counts of replicates in the 100-500 range, robust BS under ML inference becomes computationally practical for most datasets. The complete test suite is available at http://lcbb.epfl.ch/BS.tar.bz2, and BS with our stopping criteria is included in the latest release of RAxML v7.2.5, available at http://wwwkramer.in.tum.de/exelixis/software.html.