Well, here's what happened to a friend of mine:
Bright and early the morning after moving into a new house, Gary strolled out to the kitchen to start the coffeemaker and found a big grey mouse standing up on its back legs in the middle of the kitchen floor. The mouse was completely unintimidated -- just stood there and stared Gary down.
Gary stared back, pondering how to handle this little intruder. Finally, it occurred to him: Hey, I have a cat!
So Gary went back to the bedroom, gathered up his sleepy black longhaired boycat, and unceremoniously dumped him on the floor a few feet from the mouse. "There you go, cat!" said Gary, pointing. "Natural prey! Do your thing!"
The cat blinked a few times, made a half-hearted loop around the thoroughly-unimpressed mouse, and padded lazily back to bed.
So I think the answer to your question is probably no, darn it.
And the moral of the story is: Cats do not work
for their keep. Especially not at six o'clock in the morning!