Yes, but...
We have always had cats. In NM our cats have brought live mice in from the outside and failed to contain them or kill them. Our house is over 60 years old and has lots of mouse havens. Once they get behind the stove or into some other cabinet, they are quite protected. Their droppings are our only clue to their existence. Capture traps with peanut butter as bait works well and we return them to their wild habitat behind our backyard wall or into the cactus den region of our local roadrunners. Survival of the fittest!
You... let them go?
Mice are
vermin. They're not endangered, so why not just kill them?
Regular mouse traps kill the mouse more or less instantly, glue traps glue them to a pad until they starve to death.
Google image search them, if you dare.
The thing with glue traps is that they can catch things that are not mice, like birds, inquisitive children, and puppies/kittens.
It tends to be fatal for kittens, for the same reason it is for birds and mice. Glue traps should be illegal everywhere (they are in some places).
We had a mouse population explosion in 1993, after City Council, in its Divine Stupidity, decided not to spend money keeping the city boulevards and ditches mowed. The grass grew, and the mouse population loved it. They had lots more safe habitat, and roamed all over... especially in neighborhoods near this wonderful habitat, like ours (close to the river, within a 20-minute walk of the wildlife sanctuary where I used to work).
I'd hear squeaking in the cupboards, found droppings in the cupboards, holes chewed in packaging... we had to throw out a lot of food. Traps didn't work much, and my grandmother was afraid the cats would get caught in them (we had 3 cats at the time; this was a few months before we adopted Gussy).
Maggie did well in catching mice outside (sometimes evidence was left as a corpse on the back porch, but she ate most of them). Inside? I woke up one night to hear odd noises in the kitchen, so I got up to find Tomtat playing with a mouse. He batted it around, flipped it over to his mom (Lightning), who daintily took a step back. I could tell the mouse was dead by then.
I asked Tomtat, "Did you kill it?" and he got a sudden look on his face like 'uh-oh, am I in trouble?'... so I picked him up and gave him a big hug and told him what a WONDERFUL cat he was, and I was so pleased.
Then I realized that this was his first mouse kill, after
years of Maggie trying and failing to teach him to hunt. (these three were 3 generations of the same cat family - Maggie, daughter Lightning, and grandson Tomtat) So I made sure to let him know that he'd done a GOOD thing and I was very happy about it.
I also realized that nobody would believe me if I'd told them Tomtat had killed a mouse, and of course I had to do something with it. So I found one of the clean jam jars, lifted the mouse in with the tongs, and put the lid on the jar.
Then I woke up my dad and grandmother, shoving the jar with the dead mouse at them: "See what Tomtat just did? Isn't it WONDERFUL?"
My grandmother screamed a bit, my dad took a moment to realize what I was talking about... Come on, so what if this was around 4:30 am? The point is that we now had an indoor mouser!
Tomtat caught several more mice that summer. The funniest was when Lightning caught one behind the couch, carried it gently in her mouth, and then let it go when she saw Tomtat and I were there. The mouse tried to hide under the table, so Tomtat headed it off one way, and I told my grandmother to get an empty jar and the tongs. By this time Lightning had jumped up on a chair and my grandmother was freaking out: "Catch it, kill it! But don't touch it!"
I used the tongs to grab the mouse by the tail and put it in the jar, and I got the lid on. Tomtat was miffed that he didn't get to play with it, and then I realized it was still alive. How could I put a still-living mouse in the garbage?
Well, I made sure it was dead first. Then it ended up in the garbage. And the next mouse Tomtat caught, he grabbed it before I could get it, and the expression on his face couldn't be plainer: "I'm not done with it yet!"
He grabbed the mouse, took it down to the basement, played the cat equivalent of handball with it for a few minutes (dying and dead mice make a kind of squishy noise when thrown against cement walls), and when things got quiet, I went down to check. It was definitely dead, and I asked Tomtat, "Can I have it now?"
He gave me a look that was "Yeah, dunno why you like them dead, but okay," so I disposed of it.
There were a lot of complaints to City Hall over this from lots of people with mouse problems, and the next year they went back to regular boulevard and ditch maintenance.