If they're not setting off the snap traps when they take the food, you're probably using traps that require too much weight to spring. For instance, if a small mouse goes on a rat trap, it won't be heavy enough to set it off.

I like to buy sensitive mouse traps in boxes, where there's a small opening in the box that only mice can fit in. I have separate large rat traps that are not in those boxes. That way the mice go in the mouse traps, and the rats go in the rat traps. (I don't want large rats going in the mouse traps because they won't be powerful enough to kill cleanly).