My cat usually sleeps most of the day (I'm not exactly sure about the night).
How many hours a day (approximately) is it normal for a cat to sleep?
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Cats can, and will, sleep for 15 to 20 hours a day, but it's important to realize that it's usually a light doze and not the full-blown deep sleep. They're quite able to spring into action from this dozing state with very little hesitation. They do have deep sleep, which only lasts minutes, but that will happen a various times during periods of dozing.
Most cats also sleep during the daylight hours, becoming more active in the evening. As they're predators, even after so many centuries of domestication, cats follow that same pattern you see with their larger cousins: spending a lot of time in a light doze conserving their energy for hunting and then using that energy in the evening hours to hunt.
I'm entirely sure your cat is normal. :)