Low p-type doping is a limiting factor to increase CdTe thin-film solar-cell efficiency toward the theoretical Shockley-Queisser limit of 33%. Previous calculations predict relatively high ionization energies for group-V acceptors (P, As, and Sb), and they are plagued by self-compensation, forming AX centers, severely limiting hole concentration. However, recent experiments on CdTe single crystals indicate a much more favorable scenario, where P, As, and Sb behave as shallow acceptors. Using hybrid functional calculations, we solve this puzzle by showing that the ionization energies significantly decrease with the supercell size. When including the effects of spin-orbit coupling and extrapolating the results to the dilute limit, we find these impurities behave as hydrogenic-like shallow acceptors, and AX centers are unstable and do not limit p-type doping. We address the differences between our results and previous theoretical predictions and show that our ionization energies predict hole concentrations that agree with recent temperature-dependent Hall measurements.