Guitar tech people: Intonation and general string set up

I've recently sorted the intonation on my electric guitar. I've also set the bridge to a height where the strings don't buzz on other frets while playing them. Because the bridge is so high it's thrown everything off at the top of the neck when I play cregular chords, like E major, because the strings are being so hevily bent in order to reach the fretboard. Obviously, I can't play out of tune but do I have to resign myself to some fret buzz or is there something else I can do?