The identification of defect levels from photoluminescence spectroscopy is a useful but challenging task. Density-functional theory (DFT) is a highly valuable tool to this aim. However, the semilocal approximations of DFT that are affected by a band gap underestimation are not reliable to evaluate defect properties, such as charge transition levels. It is now established that hybrid functional approximations to DFT improve the defect description in semiconductors. Here we demonstrate that the use of hybrid functionals systematically stabilizes donor defect states in the lower part of the band gap for many defects, impurities or vacancies, in III-V and in II-VI semiconductors, even though these defects are usually considered as acceptors. These donor defect states are a very general feature and, to the best of our knowledge, have been overlooked in previous studies. The states we identify here may challenge the older assignments to photoluminescent peaks. Though appealing to screen quickly through the possible stable charge states of a defect, semilocal approximations should not be trusted for that purpose.