Yes, cats can and do sometimes sleep with their eyes open. But, then again, so do people sometimes but much more rarely. For people, the condition is known as nocturnal lagophthalmos. The difference between cats and people is that cats can sleep with their eyes open normally and naturally whereas for people it is normally due to a specific cause such as weak eyelids or nerve problems. Also, cats don’t sleep with their eyes wide open. ‘Sleeping with eyes open’ in this instance means with eyes partially open and with the third eyelid partially drawn across the eye.
The domestic cat has the benefit of the third eyelid which is called the nictitating membrane. When a cat sleeps with partially open eyes you see the third eyelid partially drawn over a third of the eye as shown in the photograph below from Diane Greene of her cat Liam. One eye is closed and the other partially open with the third eyelid drawn across part of the eyeball. This is not an untypical method of sleeping for the domestic cat.
Diane also publishes a photograph of Draco on the Quora.com website. His eyes are open and he’s fast asleep. You can see the corner of the eye covered with the third eyelids.
My thanks to Diane. Why don’t cats sleep with their eyes close like normal people?! There’s nothing in the literature that I have, either in books or on Google Scholar which tells me why domestic cats are predisposed to sleeping like this. It looks entirely normal for a domestic cat. Not all cats do it. I don’t have the percentage who do. I think you will find that when domestic cats sleep with their eyes open, they are normally slightly open. You won’t see a domestic cat sleeping with their eyes wide open. This may provide the answer as to why cats do it.
Their eyes are sufficiently shut to block out the light especially in conjunction with the third eyelid which moves across partially. The combination is to block out the light sufficiently to assist in sleep and to protect the eyeball. The third eyelid as an important cleansing and lubricating function which compensates the cat for a failure to blink sufficiently frequently. And in this instance, it is the key factor which allows cats to sleep with eyes partially open.
Both cat and human normally sleep with eyes closed to protect the eye and prevent it drying out. Eyes closed also makes it dark which assists in sleep.
There’s nothing more to say on the topic because there is no hard science about the reason behind cat sleeping with their eyes slightly open. There may be no particular reason for it other than they don’t need to be fully closed for the reason stated above.