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I'm sure they are composed of at least 2 drivers. I can't tell exactly how many because it's the kind with a cloth cover.
Can you get back off to get inside? Because, if the speakers don't come apart, they can't be repaired.
I can think of a few possibilities...
1. It's probably a loose connection inside the speaker cabinet. Usually, you can find the loose connection by wiggling the wires around. If the connections are soldered, they shouldn't come loose. If the connections are not soldered "quick-connects" are usually used (
picture). These might be used to make the connections between the drivers and the crossover, and/or between the crossover and the terminal-panel on the back of the speaker. Sometimes you can get a bad connection with these, and then it's just a matter of un-plugging and re-plugging in the quick-connect. It's usually some slight corrosion, and un-plugging and re-plugging it in is usually enough to clean the connection. Or, you can clean the connectors by scraping them with an X-Acto knife or something like that. (You can buy contact-cleaner spray, but usually you need some "mechanical" cleaning. )
If you can make the tweeter "kick-in" by turning-up the volume, it's most-likely some corrosion at a connector/connection, but it still could be something else...
2. The tweeter might be "getting flakey". You can get a replacement tweeter from places like
Parts Express. Tweeter prices are all over the map... You can get one for a few dolars, or a few hundred dollars.
Unless you get an exact replacement tweeter from the original speaker manufacturer, it will have different sound characteristics. It might be "brighter" or "duller" than the original. This is probably acceptable unless you have a very good high-end speaker system. But, you'd need to change the tweeters in both speakers so that they both sound alike.
Replacing the tweeter may be easy, or it might be hard. You'll need to make electrical connections. If the existing speaker has quick-connects, and the replacement tweeter has matching terminals, you're in luck. Otherwise, you'll have to solder the connections or get some quick-connects and a crimper (to crimp the quick-connects onto the wire).
If you can't find a tweeter that's exactly the same size as the older one, you might have to modify the cabinet or make some sort of adapter plate. "Traditional" wood speaker cabinets are usually easy to modify, if you're handy with wood tools. But it depends on the construction, and if you can't even get the speaker cloth off, this one might be difficult. (It wouldn't be that unusual if you had to replace the speaker cloth after modifying the speaker.)
3. A capacitor in your crossover network may be "getting flakey". Capacitors are fairly cheap, but they are usually soldered onto a circuit board. So, if you've never soldered before this might not be a DIY project. It could also be a bad connection (i.e. a bad solder joint) on the crossover printed circuit board. And, the capacitor has to be the correct value.
4. If the tweeter only cuts-out when you play it loud, the speaker may have a self-resetting circuit breaker or resettable solid-state fuse. (I think you'd only find something like that in a higher-end speaker system.) In that case, there is nothing wrong with the speaker, you are just overdriving it. And, you can overdrive a tweeter with a small (low power) amplifier because if you drive the amp into clipping, you generate "high power" harmonics. (It doesn't take much power to burn-out a tweeter... A tweeter rated at "20 watts" can't really take 20 watts.... It can take the high-frequency
part of a 20 watt signal.)
... There's really nothing else inside a (passive) speaker... One or more drivers (speakers), maybe a crossover network, and some wires.