You should never have blood, bones, tendons,and the like in your traps. Your traps should never cause that type of damage. I agree with some of the other ideas, either someone is checking your traps or you are catching animals that coyotes or dogs are eating.
If your pan tension is two pounds or higher then you shouldn't be catching rabbits. If it's less then turn it up some.
You should be able to spot a footprint in the catch circle if someone is checking them for you.
It is possible you are having toe catches. Could be caused by digging due to improper bedding or a scent issue. If you are sure scent is not a problem then make sure your trap is bedded solid. A clean trap bedded solid with more than 2lbs of tension that is positioned a couple inches lower than ground level should not have toe catch issues.
The #3 bridger is a fine trap and it is not the problem. I would run laminations on the jaws to reduce foot damage because they are a strong trap. Good luck in finding and correcting the source of your problem.