Not often. Mice frequently move into a hive because it is warm and because there is a lot of food inside. But the food the mice are looking for is the honey and pollen, not the bees. Mice are certainly capable of ruining the comb, eating the honey, and causing the colony to starve, but they are not pernicious meat-eaters. Yes, they will eat invertebrates on occasion, but a colony of bees is not their first choice.
Sometimes the bees will do in the mice. I took the following photo several years ago in my top-bar hive. Nothing is left but the skeleton of the mouse, but bees are vegetarians too. They probably stung the mouse to death and then removed all the parts they could and dumped them outside. Occasionally, honey bees will enshrine a dead mouse in propolis instead of carting it away. The propolis covering prevents bacteria from spreading throughout the hive.