We propose tests for main and simple treatment effects, time effects, as well as treatment by time interactions in possibly high-dimensional multigroup repeated measures designs. The proposed inference procedures extend the work by Brunner et al. (2012) from two to several treatment groups and remain valid for unbalanced data and under unequal covariance matrices. In addition to showing consistency when sample size and dimension tend to infinity at the same rate, we provide finite sample approximations and evaluate their performance in a simulation study, demonstrating better maintenance of the nominal α-level than the popular Box-Greenhouse-Geisser and Huynh-Feldt methods, and a gain in power for informatively increasing dimension. Application is illustrated using electroencephalography (EEG) data from a neurological study involving patients with Alzheimer's disease and other cognitive impairments.