Why don't two neutrons with the different quantum numbers pass through one another? I have read that in certain nuclear reactions neutrons can knock neutrons out of the nucleus. I know this sounds weird but why don't the neutrons pass through one another? My understanding of why things don't pass through each other (at least for charged fermions) is electromagnetic repulsion and the Pauli Exclusion Principle. A neutron doesn't have charge so the only thing that prevents them from passing through one another is the Pauli Exclusion Principle. If two neutrons have opposite spins and other different quantum numbers, then what prevents them from passing through one another/occupying the same position in space? Is it due to the neutron substructure? Why must one knock the other one out?
*Obviously this emphasizes the particle aspect over eave aspect and this question assumes that position is known withon fairly reasonable bounds.
 A: You say 

My understanding of why things don't pass through each other (at least for charged fermions) is electromagnetic repulsion and the Pauli Exclusion Principle. A neutron doesn't have charge so the only thing that prevents them from passing through one another is the Pauli Exclusion Principle.

But this understanding is not correct: you've forgotten that neutrons participate in the strong interaction.  The strong force has a "long-range" attractive component with a typical length scale of a few femtometers, and a shorter-range "hard core repulsion" which prevents two nucleons from approaching each other too closely regardless of their quantum numbers.
If you wanted to test this, you would expect cross sections for nucleon-knockout processes to be roughly the same for neutrons and protons at the same energy (since the electric repulsion is quite feeble compared to the strong force) and for both of those cross sections to be larger than for electron-induced nucleon-knockout processes at the same energy. I haven't dug into the data to verify, though, and I can think of several caveats you might need to keep in mind.
