I think it depends on the cat.
Twink was an absolute brat about chewing threads on the machine - especially the serger. Even when I covered the machine, he did his best to get to the threads and chew them.
Bear has never bothered any of the thread. He does like to "supervise" when I'm using the rotary cutter, though, so I have to watch for paws. And he likes to lay on the fabric as I'm trying to sew it.