Yep. Very easy for the squeeze on points to cut through the serving, even when they're put on right. That's why I use tie-on nocks as well, or masking tape wound on the string as nock sets. It's more durable than one would think, and can be easily taken off and new tape wound on when it wears or when you want to change the nock point height.
Way back when I did use squeeze on nock sets, I opened them enough to move or remove with a thin screw driver. A nock-set pliers, with a rounded jaw indent to accommodate the nock set, also helps squeeze them on uniformly with not too much pressure.
BTW, if you haven't cut any of the main strands on your old string, it is salvageable. Just reserve it. Quite easy to do.