I think we underestimate our 'friends' the birds ;-) This year I replanted most of my sets 40 times or more, I think I have to get to like netting for the first period until they are set properly.

have you tried to lift any plant from the ground (usually we don't but try to look for it when you harvest) ? chances are you see (a) worm(s) between the roots, maybe they know this by now and lift sets until they become rooted and harder to pull; for the worms, they hardly touch the onion.
Maybe I'm overestimating the birds now, after all the pulled all my garlic labels this year as well (I finally found the perfect permanent marker, now this ;-)