We analyze the low temperature photoluminescence properties of two multi-layer stacking of GaN/AlN quantum dots. We report drastic differences of linewidths between continuous wave and time-resolved photoluminescence experiments. After the pulsed excitation, time-resolved photoluminescence reveals a substantial red shift of the line, which keeps a fairly constant width. In continuous wave experiments, the screening of internal electric fields by accumulation of e-h pairs in quantum dot planes induces a blue-shift. For samples with large number of quantum dot planes an unexpected narrowing of the emission line is observed when the laser intensity is increased. We assign the observed behaviors of energies and linewidths to the contributions of the in-depth decrease of the degree of excitation of the different planes. Our interpretation is supported by the use of a model based on a self-consistent solution of the Schrodinger and Poisson equations within the envelope function approximation.