I've heard that cats bring home live mice in an attempt to teach us to
hunt. They feel sorry for our pathetic, obviously under-developed
abilities to snag our own food (or even to recognize freshly caught
mice as food, although that apparently doesn't apply to everyone on
this NG) and, as they would for kittens, try to help out by bringing
prey for us to work with.
Incidentally, it has been proven that well-fed cats actually catch
more mice, because the cats are in better condition, more energetic,
better athletes, etc. It's counterproductive to any mouse-eradication
goals to not feed a cat you hope will catch mice. If you've ever seen
a picture of the famous Towser referenced earlier, he was one pudgy
Also, it's absolutely true that some cats catch mice and some don't.
It's handed down from parents to kittens. Those trained as kittens
(see first paragraph) become good mousers, those that aren't, don't.
Although I have two excellent mousers now (and a couple dogs who'll
pitch in, as well), I once had a cat that wouldn't even chase a mouse,
much less catch it.