Cats do snore when they sleep. Cats are thinking animals; when they are sound asleep, they not only snore but sometimes they also talk in their sleep. When cats feel comfortable and happy around their owners, they will make a purring sound similar to purring, but sometimes it is not purring but a happy sound. Some cats are born snoring because they have short nasal passages. 
Obese cats are prone to snoring while sleeping, so it's important to monitor their weight. Cats with shorter nasal passages, such as Garfield cats, also snore while sleeping, which is normal and nothing to worry about. However, if your cat snores while awake, it may have an upper respiratory infection, and it's recommended to take it to the vet.
