1) Is any of the wiring done with push on connectors? If so, get rid of them, worst bit of junk afflicting lots of speakers. Solder everything, if you can't, get the best contact enhancer you can lay your hands on and do every connection
2) The capacitor(s) to the tweeter may be old and "lazy". Replace them with your or someone else's favourite flavour of brand
3) It may be as simple as getting the cobwebs out of the tweeter and its part of the crossover. Drive the system as hard as you dare with clean sound with lots of treble content, for hours on end. Personally, I do it with good time rock and roll, where the drummer is hammering the cymbals constantly -- I like Status Quo myself ... :-)
Frank