If the tweeter cuts in occasionally for a second or two, and if the sound from the tweeter sounds OK during that time, then the tweeter is probably OK. Sounds like a loose or intermittent connection to me. I would remove the tweeter from the baffle and make sure that there is a good connection to the tweeter terminals. Make sure that the wire isn't coming out of the crimped lug that connects to the speaker terminal. If all is nice and tight there, you can remove the bottom woofer and get to the crossover board to check the wiring at that end. I believe that's where Jim mounts the crossover board. Should everything check out and the problem still remains, I would swap the tweeters between the cabinets to see if the problem follows the tweeter or stays with the cabinet. As a matter of fact, since the tweeters are easier to handle due to their light weight, I would swap the tweeters before removing the bottom woofer. That way if the problem follows the tweeter, then you didn't have to remove the heavier woofer and break its seal.
Hope this helps,
Ron