Yes.
You thought incorrectly, as others have already commented.
There is a maximum mass that a neutron star can have, yes. However, the reason why is not quite what you appear to think. Pressure in an object like a neutron star is not applied inward; it's applied outward. It's what holds the neutron star up against its own gravity. But, as can be shown from the Einstein Field Equation, and as is confirmed by our observational data on neutron stars, pressure itself gravitates; it is part of the source of gravity. So as a neutron star gets more massive, and requires more pressure to hold itself up against its own gravity, its own gravity gets stronger, by more than you would expect just from the increased mass alone.
In addition, as the mass of a neutron star goes up and the pressure required to hold it up against its own gravity rises, the neutrons become relativistic. This reduces the ability of the neutron star matter to hold itself up against gravity with degeneracy pressure, compared to non-relativistic neutrons. (The technical terminology is that the adiabatic index of a Fermi gas--any body of matter made up of degenerate fermions--decreases from 5/3 in the non-relativistic regime to 4/3 in the relativistic regime.) The combination of these two factors--pressure gravitates, and neutrons become relativistic as degeneracy pressure rises--combine to determine a maximum mass for neutron stars, above which no amount of pressure can hold the star up against its own gravity, and it collapses to a black hole.