The concept of neutron star maximum mass is revisited. In particular we show that when the dynamical processes occuring in the first few seconds after the neutron star birth are considered, the concept of neutron star maximum mass, as introduced by Oppenheimer and Volkoff, is partially inadequate. We show that both the maximum mass concept and the final stages of the evolution of massive stars depend on the composition of the neutron star material. In particular, we find two different scenarios depending on the absence or presence of negatively charged hadrons among the constituents. In the first scenario, we show that the Oppenheimer Volkoff mass M(OV) does not represent the boundary between the value of the masses of neutron stars and black holes. In fact, we find a mass range in which both a neutron star and a black hole may exist. In the second scenario we show that, contrary to the standard view, it is possible to have a supernova explosion (accompained by nucleosynthesis and neutrino emission) followed by the delayed formation of a black hole. The latter mechamism could explain the lack of any observational evidence for a neutron star in the remnant of the supernova 1987 A.