What does a bad tweeter sound like?

If it is louder than normal, check your crossover. My experience has been either a crackling distortion or at certain notes you get a break up in the sound.

Whenever I "question" a drivers performance or simply want to conduct my own "test", I find that a 20 - 20Khz audio sweep is useful.
https://www.youtube.com/watch?v=PAsMlDptjx8

I've had crossover caps start leaking real bad and it sounded very much like the driver was blown, but only at very specific frequencies. It was also slightly louder. I was dealing with an AMT though. 

Thanks all....I just removed the tweeter from my Vandersteen 5A (left side) and am sending it back to Vandersteen for a rebuild.  I hear just as you describe...louder on that speaker, fuzzy and metallic on some frequencies.   I thought it was the midrange, speaker, but removed that one last month, got it back rebuilt, and the noise is still there.  I hope its the tweeter.

I'm not familiar with those speakers, but if the crossover isn't a real bitch to remove checking those caps out wouldn't be a bad idea. I doubt you have a leakage tester, but all the caps should measure up into the megaohms on a resistance test after a few seconds. Caps also tend to drift out of spec when the go leaky, but not always. Most of the leaky ones I've dealt with had drifted outside tolerance, but not all. They all failed to effectively block a few volts on a ohm meter though.