Summary
In a Newtonian scheme, an increasing "weight" would be supported by ideal degeneracy pressure. An increased density leads to increased degeneracy pressure (provided by the PEP) and this can simply continue until an infinite density is approached (though at a finite mass).
In General Relativity this does not happen. Instability is reached at a finite density, because in GR, the increasing density and pressure contribute to the increasing curvature of space. Thus the increase in pressure ultimately becomes self-defeating and actually triggers the collapse.
There is no question of the PEP being "overcome" or dis-applying. The maximum neutron momentum and hence neutron degeneracy pressure, will continue to increase whilst the collapse proceeds but never by enough to re-establish an equilibrium.
Details
Your book is incorrect on a number of points - possibly for simplicity.
Neutron stars do indeed consist mainly of neutrons. They are packed together at separations of around a femto-metre and at these densities, the Pauli-Exclusion-Principle (PEP) does come into consideration It is not possible for the neutrons to be all placed in quantum states with low momentum whilst also being closely packed. Since the neutrons have a distribution of quite significant momenta, with speeds getting on for a good fraction of the speed of light, they do of course exert a degeneracy pressure.
However, degeneracy pressure is not what is (mostly) responsible for supporting neutron stars. It has been known since the late 1930s (Oppenheimer & Volkhoff 1939), that ideal neutron degeneracy pressure is only capable of supporting stars up to around 0.7 solar masses. All of the neutron stars where measurements have been performed have masses greater than 1.15 solar masses.
Neutron stars would not exist without the repulsive force between neutrons provided by the strong nuclear force at small nucleon separations. This hardens the equation of state in dense, neutron-rich matter, and is responsible for the support of neutron stars more massive than 0.7 solar masses and for the "core-bounce" that is ultimately responsible for core-collapse supernovae explosions.
OK, but even disregarding this important point, your question still remains - even in the approximation of ideal degeneracy pressure, there is a finite limit to the mass and density of a neutron star that can be built in general relativistic conditions. To see why, one examines the Tolman-Oppenheimer-Volkhoff (TOV) equation for hydrostatic equilibrium, which is the correct formulation under General Relativistic conditions:
$$\frac{dP}{dr} = -\left(\frac{Gm(r) \rho}{r^2}\right)\, \frac{(1 + P/\rho c^{2})(1 + 4\pi
r^{3}P/m(r)c^{2})}
{(1 - 2Gm(r)/rc^{2})},$$
where $m(r)$ is the mass contained within radius $r$ and $P$ and $\rho$ are the pressure and density at radius $r$.
The TOV equation reverts to its Newtonian approximation when $P \ll \rho c^2$ and when $Gm(r)/rc^2 \ll 1$, which is equivalent to saying that the pressure is (relatively) small and the curvature of space given by the Schwarzschild metric can also be ignored.
The difference between the Newtonian and TOV formulations is that pressure appears on the right hand side of the TOV equation and the General Relativistic corrections act to increase the required pressure gradient. This means that increasing the pressure, by increasing the density ultimately becomes self-defeating because the required pressure gradient is increased even more. Detailed calculations (e.g. see section 9.5 of Black Holes, White Dwarfs and Neutron Stars by Shapiro & Teukolsky) show that whereas in a Newtonian scheme the density and degeneracy pressure can just keep increasing until instability is only reached at infinite density (exactly as you suppose in your question), in General Relativity the instability is reached at a finite density where increasing pressure cannot supply the increasing pressure gradient demanded by the TOV equation.
At what density this threshold is reached depends on the exact equation of state (relationship between pressure and density) governing the neutron star matter and the composition of the neutron star core. But even for the "hardest" equations of state the limit is reached at a little over 3 solar masses.
