Quite the opposite. The Pauli exclusion principle is precisely the reason that neutron stars form. Recall that in ordinary matter, a set of n spheres form the first of the quantum numbers, and the spin eigenvalues form the 4th. In general, most of these n-spheres will not be filled with electrons, but as the density increases in a forming neutron star (where electrons will be stripped from parent atoms) the electrons are forced to occupy higher n-spheres (or principle quantum numbers, the two terms are interchangeable). This gas cannot release thermal energy because the electrons have become "locked" and therefore do not relax to lower principle quantum numbers. That is why white dwarfs are luminous. This fermionic gas is termed degenerate matter and, once a star runs out of fuel, will result in a white dwarf, but if the Chandrasekhar limit is exceeded then the star will collapse into a neutron star but the neutron star cannot collapse further because two neutrons cannot occupy the same quantum state, which is a consequence of PEP. (if the Oppenheimer limit is exceeded then a black hole would be formed).

EDIT: See Post #9

I think he was asking how the proton and electron can come together under the PEP.

Because they would be essentially the same partical.

Two electrons with the same spin, energy etc... would be indistiguishable

So two similar electrons occupying the same space would essentially be one electron.

I want to return to this thread because I'm bored and because my first explanation was far too unclear, upon rereading. The stability of any star depends on the outward pressure of fusion equilibriating the internal collapse due to gravity. Once fusion stops, the star must contract under gravity.

Begin with a star of less than 8 solar masses in the red giant or supergiant phase ( all stars leave the main sequence eventually to become red giant. If the star is >4 solar masses, it will become a red supergiant after the red giant phase) of its life. Depending on its mass, the star core when fusion stops will either be composed of helium (if the star stops fusing after the proton-proton cycle), carbon and oxygen (if it stops after the triple-alpha process), or neon and magnesium. In any case, the fusing shells around the core create tremendous pressure and temperature in the core, which is confined within the fusing shell that surrounds it, but it makes the outer layers extremely tenuous until they just leave altogether. At this point, the star is a white dwarf, but the core is still contracting under gravity once fusion stops (which, like everything else) depends on how massive the star).

The white dwarf is in the center. The tenuous gas surrounding it is a planetary nebula.

Once this occurs, the electron degeneracy pressure is the only thing which stops the continued collapse of the star, since there is no energy from outward radiation pressure. The more massive the star, the greater the inward force of contraction under gravity. For these low mass stars (the overall star <8 solar masses, and the dwarf that remains after the outer layers leave <1.44 solar masses),  the electron degeneracy can stop the continued contraction of the star. This is a consequence of Pauli exclusion. This (core <1.44 solar masses) is called the Chandrasekhar limit. The reason I mention this is because it is necessary to understand neutron star formation.

The principle quantum numbers are shown above. In the core of the white dwarf, the density is such that the electrons occupy all of the possible quantum numbers. In normal matter, this does not occur. This is called degenerate fermi gas. The outward pressure of the electrons because all quantum numbers are occupied will halt contraction only if the Chandrasekhar limit is not exceeded.

Now, begin with an arbitrary high mass (>8 Solar masses) star in the red supergiant phase of its life. All of these stars will be to go past the formation of neon and magnesium. They will be able to fuse silicon to make iron ash in their core. The nucleosynthesis process must stop here because iron is the most bound nuclei.  All of these stars above 8 solar masses undergo a supernova as part of the last stage of their life. As with any fusion reaction, the production of photons in the core during nucleosynthesis occurs. In high mass stars with an iron ash core surrounded by magnesium, neon and oxygen (surrounded by carbon, surrounded by helium, surrounded by hydrogen) the gamma radiation is sufficiently energetic to break the nuclei in the core until the star core is composed almost entirely of protons, neutrons and electrons. Once this occurs, the inward pressure on the core cannot sustain the occupation of principle quantum numbers by electrons. They must be forced into the protons (they have nowhere else to go). As a result, the core is composed wholly of neutrons This process results in a virtually instantaneous tremendous increase in density of the core. This again is a consequence of Pauli exclusion. If the core contraction is too high for the pressure from electron degeneracy due to PEP to stop the contraction, then the only place the electrons can go is into the protons. Because the Chandrasekhar limit is exceeded, the contraction is too great for electron degeneracy to halt. But the neutrons are also bound under the PEP. Two neutrons cannot occupy the same quantum state. So as the star contracts, the neutron degeneracy pressure is what halts the contraction this time, instead of electron degeneracy pressure. But this time the process does not just stabilize. The PEP repulsion of neutrons is tremendous. The outward degeneracy pressure overshoots the inward contraction, so the star expands to an equilibrium state between neutron degeneracy and gravity. The overshooting of the inward contraction by Pauli exclusion is catastrophic. A massive shockwave is sent through the star which blasts off every outer layer and leaves a tiny core of neutrons. Now we have a stable neutron star. Just for emphasis, when I say "tiny core" I mean really fucking small and dense. Imagine the sun compressed to the size of Manhattan (this is just a scale comparison. Obviously this cannot happen to the sun because  the sun is too small to exceed the Chandrasekhar limit). This whole process is a consequence of Pauli exclusion. This process is a stunning demonstration of the accuracy of predictions of quantum mechanics.

But one process that does violate Pauli exclusion, and that no physics, not even quantum mechanics, can describe, is what happens when the Oppenheimer-Volkoff limit is exceeded. This is a logical extension of what we saw above. If above a certain mass (the Chandrasekhar limit) the electron degeneracy due to PEP cannot stop inward contraction, there should be another mass limit above which the neutron degeneracy due to PEP cannot stop the inward contraction. Once gravity overpowers neutron degeneracy, there is no outward pressure which can halt its continued contraction into a black hwole. The mass limit (the core >3.2 solar masses, or the original star before everything was blasted off>40 solar masses) is called the Oppenheimer-Volkoff limit. Once the Oppenheimer-Volkoff limit is exceeded, the neutrons have nowhere to go, but their outward degeneracy pressure cannot stop the inward contraction. The star is too massive. Eventually, the mass will be forced into a volume such that the span of the radius is less than the Schwarzschild radius. Once this happens, the event horizon is formed, and a black hole is born. There is no physics which can describe the physical matter that undergoes this process. As an illustration of the mind-bending density that results when the contraction overcomes neutron degeneracy, the Schwarzschild radius of Earth is just under one tenth of one millimeter. This is the amount of space we would have to compress the mass of Earth into in order to make a black hole out of it.

Once the mass is compressed to less than the Schwarzschild radius, the bounding surface of the SR will be the event horizon of the black hole. No information can leave the event horizon, and as per gravitational time dilation, anyone far from the event horizon observing someone travelling into a black hole would simply see them stop at the event horizon (actually, they wouldn't see them at all since gravitational time dilation is equivalent to gravitational redshift, and the light would simply be redshifted to infinite wavelength).