The electron emission by micro-protrusions has been studied for over a century, but the complete explanation of the unstable behaviors and their origin remains an open issue. These systems often evolve towards vacuum breakdown, which makes experimental studies of instabilities very difficult. Modeling studies are therefore necessary. In our model, refractory metals have shown the most striking results for discontinuities or jumps recorded on the electron emitted current under high applied voltages. Herein, we provide evidence on the mechanisms responsible for the initiation of a thermal instability during the field emission from refractory metal micro-protrusions. A jump in the emission current at steady state is found beyond a threshold electric field, and it is correlated to a similar jump in temperature. These jumps are related to a transient runaway of the resistive heating that occurs after the Nottingham flux inversion. That causes the hottest region to move beneath the apex, and generates an emerging heat reflux towards the emitting surface. Two additional conditions are required to initiate the runaway. The emitter geometry must ensure a large emission area and the thermal conductivity must be high enough at high temperatures so that the heat reflux can significantly compete with the heat diffusion towards the thermostat. The whole phenomenon, that we propose to call the Nottingham Inversion Instability, can explain unexpected thermal failures and breakdowns observed with field emitters.