Uh-huh...
Yes, and a couple of my cats killed birds. Cassandra did it because surprise! hunting is instinctive for cats. She ate her kills, though, rather than leaving dead bodies lying around.
Maggie hunted birds, but in her case she was living under the back porch and nursing a litter of kittens. She hunted for survival before moving into the house permanently after having her second litter. She still hunted afterward, because instinct and I suppose she wanted to supplement the cat food. She was eating for six, after all. And a year later, she tried to teach her grandson, Tomtat, to hunt...
He was hopeless at it. His own mother never taught him to hunt. But eventually both of them did turn out to be hunters. The summer of 1993 saw a mouse invasion, and I woke up one night to hear strange noises from the kitchen. I went downstairs to find Tomtat batting a mouse around. Lightning (his mother) was watching him, and I remember how he flipped the mouse over to her, and how she backed away as it landed at her feet.
The mouse was dead by this time, and it suddenly occurred to me: This was Tomtat's first kill. I figured my dad and grandmother would never believe me without evidence, so I got an empty jam jar, put the mouse in it, and woke them up. Nothing like waking up at 4:30 am to find a jar with a dead mouse in your face... I told them, "Look what Tomtat did, isn't it WONDERFUL?"
So Maggie hunted the mice in the yard and Tomtat hunted the mice in the house. By that time Lightning had figured out hunting, but her prey was quite a bit smaller. She hunted insects. I didn't have Cassandra then; she'd died the year before, and Gussy came along in the fall of that year.
In 1994, Gussy made the connection between "I'm a cat" and "I should be hunting something". So he tried tackling a bee, which stung him. His paw swelled up to twice its normal size, and since this was on a Sunday and the vet's office was closed, I literally ran downtown to the library to look up whether bee venom was fatal to cats. Turns out it can be, if they're allergic. Fortunately Gussy wasn't, and the swelling lessened over the next few days. But following the advice in the book, I kept a close eye on him to make sure he didn't develop breathing problems.
After that, Gussy decided hunting was overrated. And then he discovered he could get high on honeysuckle, so I think the birds were pretty well safe from him.
I don't know what Chloe hunted in her first year, since I didn't get her until she was about a year old. She would have had to hunt for survival, either birds or mice, and she might have kept that up during the 5 years I had her when we still lived in the house. I have no idea.
But after moving to my first apartment, neither Chloe nor Maddy hunted. I didn't let them out. Maddy found the pigeons around here to be really annoying, and I'd happily commit violence on them myself if it wasn't illegal. Maddy wouldn't even hunt insects. She'd get mad at me for killing them, because she wanted to play with them!

The closest she ever got to hunting was her favorite toy mouse. She never killed any real ones.
My dad's cat was a mouser, and I remember one day when I found a dead mouse he'd left for me on the back porch. Some people would get mad or freak out, but I just said, "Thank you, Sammy," and disposed of it in the garbage when he wasn't looking (he really did expect me to eat the thing). Sammy ate his kills, and I wish I'd been there when my dad's girlfriend opened the door to find Sammy sitting there, wanting in. He had a dead mouse in his mouth, the tail hanging out the side.
Later on, they found out what he did with the mice after he'd finished with them. He had a mouse cemetery in the cellar, with the remains he hadn't wanted to eat all lined up in a row behind the furnace.