You may be referring to student accident policies which have nothing to do with liability insurance.

A liability insurance policy pays those sums for which the insured becomes legally liable (if there is negligence), provides a defense if there is no negligence, and gives the insurance company the right to investigate and settle if expedient.

It's possible that you have seen insurance companies paying for a child's injury with no apparent negligence. That's called go-away money and it's usually a small payment just to close the claim.