This is almost certainly an issue with the nut slots binding. Gibson guitars have a wide headstock that bends back at a steep angle. This means that the strings need to move left or right inside the nut slot and move back. This is a recipe for the strings sticking in the slots.
You could have the nut replaced by a good luthier. I'd also look into a Teflon based lubricant for the nut slots. It's the only thing that worked like a dream for me. Graphite and oil didn't help at all.
Another slight possibility is that the bridge is shifting. I bought a Les Paul Custom new in 1991, and it came from the factory with a slightly small screw holding up one side of the bridge. This allowed the bridge to suddenly shift forward and backwards, throwing the guitar out of tune. I think it took me 20 years to finally figure it out (and fix it for $20).
I hope this helps.
Mike
PS - I've been waiting for over a year for Gibson to give me clearance to post to the Talk To Henry forums. I think that's pretty crappy.