Does a school pay fully for injuries when it causes an accident?

A piano fell on me at school and I think three of my teeth are starting to die. The school's insurance company will only pay $3000 dollars but I think I need more than that to get fake teeth.
Will the school pay 100% for all of the injuries even if I get the best fake teeth available?

One cannot "cause" an accident. One can be at fault, but an accident is by nature, an accident. The party at fault can be held responsible.

If this happened at a business, you'd be able to get reimbursed for all reasonable and necessary medical; however, in many states schools are immune from being sued. In Texas several years ago, a light pole fell on a girl's head and the school paid nothing. The girl was not able to sue.

Go to a dentist and tell the dentist your situation. You are dealing in hypotheticals right now. You don't know what needs to be done regarding your teeth and you don't know if it will exceed $3,000. I doubt the school is going to pay you anything over and beyond what their insurance provides.

How long has it been since the accident? If it has been a long time, you would have difficulty even proving that the problem you are having with your teeth is related to the accident.

If your teeth are, in fact, dying, you would not have to get false teeth. You would have to have root canals and then have crowns placed on those teeth. First, however, you would have to have x-rays to determine if root canals are necessary. Figure about $200 for the x-rays, $800 each for root canals and I don't remember how much crowns are. There are porcelain and gold. It would depend on which you got and for what kind of tooth.