Context. The literature is rich in analysis and results related to thermally pulsing-asymptotic giant branch (TP-AGB) stars, but the problem of the instabilities that arise and cause the divergence of models during the late stages of their evolution is rarely addressed. Aims. We investigate the physical conditions, causes and consequences of the interruption in the calculations of massive AGB stars in the late thermally-pulsing AGB phase. Methods. We have thoroughly analysed the physical structure of a solar metallicity 8.5 M-circle dot star and described the physical conditions at the base of the convective envelope (BCE) just prior to divergence. Results. We find that the local opacity maximum caused by M-shell electrons of Fe-group elements lead to the accumulation of an energy excess, to the departure of thermal equilibrium conditions at the base of the convective envelope and, eventually, to the divergence of the computed models. For the 8.5 M-circle dot case we present in this work the divergence occurs when the envelope mass is about 2 M-circle dot. The remaining envelope masses range between somewhat less than 1 and more than 2 M-circle dot for stars with initial masses between 7 and 10 M-circle dot and, therefore, our results are relevant for the evolution and yields of super-AGB stars. If the envelope is ejected as a consequence of the instability we are considering, the occurrence of electron-capture supernovae would be avoided at solar metallicity.