There's a simple way to know for sure if it's the nut. with the nut locked, pluck the high E, and as it's ringing, press on the string behind the nut, where the headstock is. if it goes sharp, then the pad isn't locking down properly.
also, check the 2 screws that hold the nut in place. those will loosen up over time.
you can also try to switch the little blocks in the saddle to see if the same happens with another block.
And again, check the screw that holds the saddle, and make sure it's not bottoming out on the guitar body preventing it from screwing down all the way.