err.... not always.
Our cat loved to bring live mice indoors....let them go so he could taunt them. He'd finally kill them and maybe eat half of them leaving the other half so we could see the trophy.
(he's got a flapper door in the window so he can come/go as he pleases)
Not once, but TWICE I walked into the basement bathroom (finished with full shower). One of the times, he was standing by the full length glass door (which was open) and inside the shower was a live rabbit that he dragged into there. The poor rabbit was literally bouncing off the walls trying to escape. Meanwhile, the cat was at the door watching. The rabbit was bleeding and the shower walls (at rabbit height) were covered in blood. Looked like Helter Skelter in there, rabbit style.
The NEXT time..... I was walking into bathroom....he's sitting on floor in front of me with his back turned to me. As I walk in, he turns around (think Reagan in Exorcist) and has blood on his face.... in front of him is a decapitated rabbit body. Head is totally missing and the cat is eating the shoulder meat.
He has since lost ALL four of his canine teeth and can't hunt very well. Not a mouse (nor rabbit) in the house since then AND he's probably not gone outside but 6-10 times over the last 4 years (since teeth pulled). Oh, one of his canine was falling out, took to vet who took them all. We didn't just decide to pull his teeth.