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Couple more examples (I've half-written the trope already, here!)
The Fifth Element - Leeloo magically producing a gun after Korben tries to kiss her while she's asleep.
The Bone Collector - "there's a law against molesting the handicapped"
I'm fairly sure Clint Eastwood does this in one of the Spaghetti Westerns, too, as he often appears asleep under his hat, but seldom is.
Edited by 94.174.54.232Instant Waking Skills is related, as are Super-Reflexes and Kung-Fu Clairvoyance
As for feigning sleep, I think Playing Possum would cover it.

I'm looking for a movie trope I've seen many times in film, but can't find described here.
It involves someone who is apparently asleep, but displays lightning fast reactions when another character goes near them, either to wake them or to steal/take something from them.
Often characterised by the sleeping person grabbing the arm of the approaching character, demonstrating that either they've got some sort of instinctive sixth sense when they're asleep, or they never actually sleep, the better to be alert to threats.
It can also be applied to people who appear asleep but then open one eye the moment the bad guy(s) start tiptoeing around outside the room.
The third variation on this is characters who continue to pretend to be asleep for the benefit of another character, usually waiting until they leave before moving. The departing character meanwhile thinks they snuck away unobserved.
I've tried searching for these themes and can find nothing on this site to relect them.
Examples: Soloman Kane does this when the Crowthorne boy reaches out to wake him up.
Antonio Banderas does the second variation in Desperado while Salma Hayek is singing.
All of the Viking warriors do this in The 13th Warrior, even making sleeping and snoring noises while they lie awake listening to the bad guys gathering outside.
There are many other examples, although of course I can't remember many at the moment!
Am I missing this on the site, or is it insufficiently notable to justify a page?