In Florida, public hospitals and even private agents of those hospitals and medical centers, are granted protection from unlimited damages by the Florida sovereign immunity statute. This statute caps damages for public hospitals and employees (as well as all State, county, and city employees) at $100,000 (which will be raised to $200,000 in October under new legislation).
The only way to exceed this cap is to have a private bill -- called a claims bill -- passed by the Florida Legislature and signed by the Governor agreeing to a certain amount of money damages above the cap. This is, as I'm sure you can imagine, rare, and often reserved for catastrophic cases.
Additionally, Florida also has separate caps in place for pain and suffering damages in all medical malpractice cases.
This does not prevent your friend from suing, but does limit the damages available. Lawyers experienced in medical malpractice cases know that these are among the most complicated civil cases to prepare and try, difficult to win in today's climate, and therefore want to know whether their time and costs (which can literally be in the tens of thousands of dollars) are going to be paid in the event of a win at trial. When we see the specter of sovereign immunity caps, we have to consider the cost/benefit of pursuing the case.
In truly catastrophic cases, many of us will go forward, even if the case involves a public entity.
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Certain government-run hospitals can have some sort of sovereign immunity for some liability theories alleged. And, a legislature can immunize certain important functions of a hospital (teaching, research, etc...) from tort liability.
I agree with my colleague's answer and would add that the immunity does not stop you from suing, but severely limits damages. However, there have been some good case decisions in recent years that stand for the proposition that you cannot simply call something a government hospital and thereby protect all of its functions and actions, if it is truly not a government operation.