It depends on how serious the bullying is. If your child has been repeatedly assaulted while at school, you have notified the school about the problem, and the school has failed to take adequate precautions, then yes, the school district could potentially be liable for failing to act. Again, this will turn on the specific facts of the situation.
If the bullying/harassment is based on race or sexuality, that could give rise to causes of action under federal law (e.g. violation of the Equal Protection Clause). Even if the bullying is not related to race or sexuality, severe bullying and harassment could give rise to a number of state law claims. The school district potentially could have been negligent in failing to take adequate steps to provide your child with a safe school environment.
Please feel free to contact my office if you need further assistance with this. I would be happy to help in any way I can.
My response to this question is a response to a hypothetical situation based on limited facts. I am not your attorney; you are not my client and we do not have an attorney-client relationship. If you need a lawyer, you should contact one in your area. If you would like to talk with me about your case, you can call my office.