Can I sue a jail for medical neglect?

Yes, especially if this happened while you were serving time in a private jail or if you were treated by private medical contractors while incarcerated. For-profit corrections systems put quantity over quality every time; they get paid per inmate, per day. The less they spend, the higher their profits. Cutting costs means cuttings corners, and inevitably that means a lower level of medical care.

Even if a correctional facility is not privately owned and operated, its medical providers might be. Private medical contractors, like private jails and prisons, are often not subject to government regulations or oversight. They have less reason to provide adequate medical treatment.

There is a silver lining to this dark cloud: because private medical contractors are not government entities, they do not receive the same protection from lawsuits that government agencies do. It is far easier to successfully sue a private jail or the medical staff at a jail because those private companies do not have the same protections against lawsuits that government owned jails usually do.