So a thought occurred to a friend of mine the other day: in a neutron star, neutrons are prevented from sitting directly on top of one another due to the Pauli exclusion principle, what with neutrons being spin-1/2 particles. This manifests itself as a pressure that keeps the star from collapsing in on itself.
However, composite objects of half-integer spin can have integer spin. For instance, a 'dineutron' has either spin-0 or spin-1, and is hence a boson. This fact is what allows, for instance, helium-4 to undergo Bose-Einstein condensation. So merely by considering the neutrons in a neutron star in pairs, we ought to conclude that the star should collapse in on itself, on account that it is composed of bosons. Obviously this is not right, not just because it is not observed, but because how we group the particles into composites in our heads cannot affect what we actually observe...
The same holds for helium-4: if I want to think of my fundamental particle as the helium-4 atom, I conclude that helium-4 can undergo Bose-Einstein condensation. But if I think of my mixture as being a 1:1:1 collection of electrons, protons and neutrons, then the Pauli exclusion principle applied to each species prevents this condensation --- there is a 'Fermi degeneracy pressure'.
What's going on here?