I do love our family cat, but she's a born killer. Over the years, despite bells on her collar and what have you, she's brought in birds, mice and we even had one stray who brought a baby rabbit!
She's well fed etc, she just can't resist the hunt. If we saw her trying to kill something we'd normally step in and get her to stop then release the prey if it was fit enough to run or fly (mum nursed some badly beaten birds back to health!) but then again sometimes we were just too late.
Now that she's getting older she doesn't catch many birds and stays in most of the time.