Logically over a dislocated pinky any logical attorney or lawyer will laugh you out of their office, however if it were a more serious injury that was not properly acknowledged and documented by said school, there is a more viable chance. Per say you get a concussion, easier to sue over if they did the same steps but did not document any reports regarding said injury, and this is only applicable to public school districts in the U.S due to government regulation. Typically if you are an enrolled student there and are on grounds at any point, you are their legal responsibility for your health/well-being and safety.
TL;DR yes you could sue but your parents would have to sign off with an attorney and deal with that stuff, and you will get ridiculed by said attorney for even trying to pursue legal action over a dislocated pinky