According to Prof.Stassun in the Life and Death of Stars, neutron star sizes are limited by Neutron degeneracy pressure (and nuclear force) which states that the size of the star can be no greater than 3 solar masses. When the closely packed neutrons get closer, the heisenberg uncertainty principle would cause/necesitate the neutrons to move faster than the speed of light to balance the certainty in position. Instead they collapse into a black hole. Why would this collapse if caused by a star feeding it not produce a supernova just like the secondary supernova that occurs when white dwarfs are fed matter from a star greater than 1.4 solar masses but of the type produced by giant red giants that end up as black holes.
Why do neutron stars fed by another star not produce a second supernova like white dwarf supernovae?
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The reason why white dwarfs undergo supernovae when in an accretion situation is due to runaway nuclear fusion because they are unable to regulate their temperature as opposed to a normal star and so the resultant energy unbinds the white dwarf.
A neutron star in an accretion situation would just become more and more compressed until neutron degeneracy pressure is overcome and the neutron star falls into its own event horizon. A neutron star's matter is in a much lower energy configuration compared to a white dwarf (being held up by electron degeneracy pressure) and so ignition really isn't possible. Remember that white dwarfs are still composed of Carbon and Oxygen nuclei while a neutron star has no such nuclei for potential fusion or ignition.
EDIT: Time to address a very silly mistake. There are indeed nuclei on and just under the surface of a neutron star, most likely composed of Fe and other elements from the star before.
It is also possible for there to be fusion on the surface of a neutron star. If H and He accumulates to high enough amounts from a neighbor, then very brief runaway fusion (a timespan of seconds) can occur, which is detected as X-ray bursts. This is similar to the nova event of a white dwarf, and can occur periodically.
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$\begingroup$ Has there ever been an instance of a neutron star collapse being detected if detection is possible? What exactly happens that's different from the collapse of a red giant into a black hole(Which is composed of Fe all the way down to H). $\endgroup$ May 29, 2018 at 13:37
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$\begingroup$ What happens to the infalling matter on the neutron star? Is it continuously converted to neutrons, or does the neutron star accumulate a layer of non-neutronium matter? If the latter, could that layer ignite? $\endgroup$– ÅsmundJan 22, 2019 at 9:26
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$\begingroup$ Something that I was incorrect on was my statement on nuclei. On and near the surface of the neutron star, nuclei do exist. So if matter accretes onto the Neutron star, it would constitute whatever the matter was composed of. This matter would be distorted by the properties of the neutron star, but it wouldn't undergo neutronization. $\endgroup$ Jan 22, 2019 at 15:49
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1$\begingroup$ @Åsmund Hydrogen falling onto a neutron star from a companion picks up a lot of kinetic energy, since the neutron star's escape velocity is relativistic (Wikipedia says from a third to half of lightspeed). So that hydrogen can easily undergo fusion when it hits the neutron star, but it's unlikely to induce fusion in the heavy atoms on the neutron star's surface, since that requires outrageous amounts of energy that you only get in a supernova collapse or a neutron star collision. So that's very different to what happens when a white dwarf accretes matter from a companion. $\endgroup$– PM 2RingJan 22, 2019 at 20:50
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$\begingroup$ There's a nice explanation of neutron star structure, with a diagram, in this Astronomy answer by Rob Jeffries. $\endgroup$– PM 2RingJan 22, 2019 at 20:59