I'll be the first to admit, my cat Punkie has to sit next to my machine and watch every move I make while I'm stitching. Fortunately, she never bothers my thread, but I still take a few precautions.
First, I keep a thread catcher at my side and make sure every piece of thread goes inside. Even with that, I still back away from my machine occasionally and look to make sure I didn't drop a piece.
Secondly, even if I get up to grab a cup of tea or leave my room for anything, I pick up my thread catcher and put it up where she can't get it while I'm gone. She's old, so she can't jump high.
Thirdly, after my sewing session is complete for the time being, I scan the entire area really well to make sure I didn't leave anything behind on the chair, table, machine, carpet, and myself.
I just wanted to make you aware of this hidden danger that many people don't know about. And if by chance you see a string of anything trailing behind your pet (you know what I mean here) DO NOT PULL OR CUT IT OFF. This is an EMERGENCY and you need to run to the nearest animal clinic you can get to. If it is very long and the animal is trying to tug or bite at it, you can trim it, but you need to leave some out for the vet's examination.