We examine the response of a pulse pumped quantum dot laser both experimentally and numerically. As the maximum of the pump pulse comes closer to the excited-state threshold, the output pulse shape becomes unstable and leads to dropouts. We conjecture that these instabilities result from an increase of the linewidth enhancement factor a as the pump parameter comes close to the excitated state threshold. In order to analyze the dynamical mechanism of the dropout, we consider two cases for which the laser exhibits either a jump to a different single mode or a jump to fast intensity oscillations. The origin of these two instabilities is clarified by a combined analytical and numerical bifurcation diagram of the steady state intensity modes.