Photoluminescence and high-resolution selective excitation of luminescence have been used to study shallow donors in non-intentionally-doped Te-rich or nearly stoichiometric CdTe crystals and their evolution after controlled thermal annealings (specifically under Cd overpressure). It has been shown that there are at least six native donors. For each of these donors, the (1s\ensuremath{\rightarrow}2s) and (1s\ensuremath{\rightarrow}3s) transition energies, the ionization energy, and the exciton localization energy have been determined. The ionization energies range from 13.71 to 14.79 meV and the localization energies obey Haynes's rule. Doping experiments clearly identify the chemical nature of three donors (Ga, In, and Cl), and two other native donors are tentatively ascribed to Al and F. This study clearly indicates that the conversion to n-type behavior induced by the annealing under Cd overpressure does not involve stoichiometric defects like interstitial Cd but is essentially related to a modification of the solubility of the residual contaminants of the ingots.