The short answer is, it depends. Wait, you might think, "aren't school entrusted to protect my child? How can they evade responsibility?" You are right; schools are required to care for the students while they are under the care of the school. But the law also recognizes that the school cannot be liable for every injury sustained on campus because of circumstances beyond the school's control.
For example, the most frequent type of school-related injuries comes from sports. While sports are responsible for the clear majority of injuries, they compose a very small sliver of the personal injury suits. The problem for sport-related injuries is that the child assumes the risk of injury when he or she joins the sport. Furthermore, most teams now require students to sign a waiver of liability. But keep in mind, that waivers do not waive the duty of coaches to behave reasonably, including taking proper care of the students.
Another common cause of injury relates to bullying, fights, and shootings. Schools can be held liable if they allow bullying or fights to continue. The operative issue in these suits is showing that the school was aware or should have been aware and failed to act reasonable. A reasonable action is almost always ending the fight and separating the children. But schools are also not equipped for all-day, 100 percent coverage. Therefore, there are limits.
But in the situations of well-known bullies and victims, schools are required to intervene. But, the school mostly retain autonomy is determining how best to intervene, provided it is reasonable.
If your child was injured while on school grounds, you should contact a lawyer at your earliest convenience. As you can see, an injury, even a severe one, does not necessarily give rise to a valid claim for compensation. A lawyer can investigate your child's injury, including the circumstances surrounding it, and advise you of the best way to pay for your child's injuries (be it through health insurance or a lawsuit). You don't need to figure it out on your own; a lawyer can help.