A recent paper in Neurology used statistical techniques to investigate the integrity of the randomization in 33 clinical trials conducted by a group of investigators. Without justification, the approach assumed that there would be no impact of correlation among baseline variables. We investigated the impact of correlation on the conclusions of the approach in several large-scale simulation studies that replicated the sample sizes and baseline variables of the clinical trials in question and utilized proper randomization. Additionally, we considered scenarios with larger numbers of baseline variables. We found that, with even moderate correlation, there can be substantial inflation of the type I error of statistical tests of randomization integrity. This is also the case under no correlation, in the presence of some discrete baseline variables, with a large number of variables. Thus, statistical techniques for assessing randomization integrity should be applied with extreme caution given that very low p-values, which are taken as evidence against valid randomization, can arise even in the case of valid randomization, in the presence of correlation. More generally, the use of tests of goodness of fit to uniformity for the purpose of testing a global null hypothesis is not advisable in the presence of correlation.