Comon sense would lead me to think that yes cotton would break down as compost but would take longer than some other things.
I am always leary of putting strips and threads out where something might swallow them and it would get caught on their insides. But, I used a thin piece of baby yarn to tie up a clematis vine and within two days "something", most likely bird or squirrel had cut it into pieces and taken off with some of them. I found that amusing.