The reason "neutrons don't overlap", as DarenW explained it, has to do with intricate forces at play that take into account the spins, iso-spins and symmetry of the wavefunctions.

However, assume I have two paired-up neutrons. I assume they should behave as bosons now. What prevents them from overlapping? Are the same forces (that are beyond my current knowledge) in play?

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    $\begingroup$ "I assume they should behave as bosons now" Compound systems can behave as units as long as they can remain coherent (see, for instance superconductivity), but you can always write the wave function of the compound in terms of it's constituents and it still has to obey the apposite symmetry which is in essence why nuclei get bigger as you add nucleons even though we know the bits are pairing up at some level (search on "short range correlations"). $\endgroup$ – dmckee --- ex-moderator kitten Apr 3 '13 at 17:13

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