Most-likely, you've got string slippage over your bridge saddles or over your nut. This is usually caused by using a bridge or a nut which doesn't lock the string. If you're using a "Fender-style" bridge (like the one shown), I don't care how "frictionless" your bridge saddles claim to be, you're going to get slippage during diving/pulling (which the "frictionless" ones are designed to do, but one or more of the strings is also going to "hang up" when trying to slide back to its original position when you release the bar.
Same thing goes for pullied contraptions at the nut...
The only exception to this may be to use bridge saddles with actual pullies in them. You can get a set of 6 for about $15 from eBay, Stewart Macdonald, GuitarFetish, etc. Just google "roller saddles guitar" and you'll find them.
As far as nuts, though, you really need to get a locking nut. I don't trust graphite or roller-style nuts very much. And there's kinda no reason not to get a locking nut. They're less than $10 on any guitar parts site or eBay. You'll probably need to chisel/carve out the recess to accept the larger frame, and you'll need to be careful to get one which is the same width as your guitar neck, but I think they're worth it.