Yes, mice and other rodents normally avoid areas with cats, and it is also true cats and some breeds of dogs make excellent mousers/ratters; however it does not always automatically follow that a home infested with vermin will be totally free of the pest by taking in the random cat or dog.
When one had a cat, it did catch one mouse just after we moved in, and never saw another again. However that does not mean the darned things weren't still around, they could have been just very careful to stay out of the cat's way. Heck "Tom" never rid the house of "Jerry", and in real life there are many stories and indeed pictures of cats happily sharing a meal and or bed with a mouse.
If a cat is not taught in kittenhood what to do with mice, instincts will only go so far. Formerly pet cats allowed to go feral will in time learn the job, but that is because they have to feed themselves. However starving an indoor cat of food in the hopes it will turn into a mouser does not work.
Indeed one of the first signs persons have of mice or other rodents is that the vermin get into bags of the pet's food or raid the food bowl for their meals.
Have been reading postings on the Interent about persons with various mice and or rat problems, and a cat though often recommended, does not always solve everyone's problem. Indeed animal shelters and feline breeders in our area will NOT allow one to take a cat or kitten if one states it is for mouse hunting. Why? Sadly because persons once having adopted the pet, can become quite angry if the feline does not do it's "job". Often this can lead to the pet being abandoned and or treated badly.
Living in an urban area, would be very careful about allowing felines or dogs (many people in NYC own various breeds of terriers, including Jack Russells, who go crazy during their walks when they catch the scent of rodents in the streets/garbarge), to deal with much less come into contact with sewer/filth living rodents. Aside from the parasites and diseases, there is a good chance the thing may have ingested poison, (NYC itself and it's residents spend fortunes on the stuff, and use it liberally). Many pets and even wildlife die each year from ingesting poisoned vermin.