Is it possible to clench your jaw so hard that your teeth shatter?

Possibly, but rare: If you have been a habitual clencher for a long time and you've worn your teeth pretty bad it may be possible to shear off pieces of enamel on the side. In general teeth are pretty strong. If your teeth have been compromised structurally with large fillings or cavities they would be easier to break. If you clench get an otc mouthguard until you can see the dentist for a more durable one.

Yes: Especially if your teeth have had dentistry done, or need dentistry done. You might consider getting a "brux-guard" to lessen the pressure that clenching puts on your teeth.

Yes: Yes. Before this happens, see dentist for moughguard.

Hardness: Diamonds and granite are much harder than teeth, but both can be broken. Avoid clenching and have a bruxism guard fabricated to help protect your teeth.

Shatter no, chip yes: The proprioceptors in your muscles prevent you from clenching hard enough to "shatter" your teeth, however you can clench hard enough to chip your teeth. Teeth that have been restored with large composite and/or amalgam restorations can possibly shatter due to clenching.

Yes: Rare to see, but it can happen. Much more frequent if teeth have larger fillings which extent from the front in between over the top and down the back in between areas. These can more easily be split. See your Dentist for an occlusal guard (Thermaguard is very comfortable) to help prevent this and get advice about reducing / eliminating the clenching / grinding habit of oral compulsion. Good Luck.

Nocturnal Bruxism: also called night time clenching, may cause tooth wear, and even damage or break teeth. See your dentist for night guard fabrication. Good luck.