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This question, to me anyways, is basically a balancing act between 2 possibly opposing effects.

Take a neutron star with just too small a mass to overcome it's degeneracy pressure, failing to collapse into a Black Hole. (Chandrasekhar limit)

Now assume it acquires a bit more material, that has a small mass but has large angular momentum.

Would this extra angular momentum provide sufficient rotation act to stop further collapse or would this additional K.E. act as an effective mass, through $mc^2$, to actually promote a collapse into a Black Hole?

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Yes, a sufficiently massive neutron star can prevent collapse to a black hole, up to a point. The most massive neutron stars are thought to be spinning very rapidly, and they exceed the mass of the theoretical limit of a non-rotating neutron star. However, most models predict that a neutron star couldn't spin much faster than 1 kHz because they would break apart or emit gravitational radiation faster than they could spin up (see here).

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