This sounds to me that you may have the bridge set slightly low. The first thing I'd try to do is to raise the bridge height just enough to stop the problem and see if that's workable for you. Although bridge height is a big part of the overall action and playing height, you may find that tweaking the relief will minimize the difference in feel from the higher height. The bridge height dominates the feel in the high registers, but for most of the rest of the neck, the feeling of string height is coming from the neck relief instead. If you play with a lot of relief (bow away from strings), then the action will feel high no matter how low you put the bridge. So setting the relief slightly lower may allow you the extra height at the bridge.
The next thing to check would be to take a look at fret wear. In a perfect world, if you fret at the 1st and 24th fret, you should see a gentle bow, reaching its greatest point in the middle (around the 12th fret). The max gap is sort of dependent on the playing feel that you like, but is usually pretty small (like a medium to heavy pick, for me). In addition to checking that, you should see the fret gaps increase and decrease linearly toward the middle. In most cases, the finished height of the fret should be about the same across the neck too, but this can be variable - if there was a high spot on the fingerboard, then having some frets ground smaller to compensate isn't wrong, even if it's non optimal. If you see irregular gaps and irregular fret sizes, then you probably will need to look more closely at that to allow low action.
That ought to at least get you started.
Good luck,
David Fung