It's unlikely (but I'm not sure what you mean by private insurance and medical, do you mean public health care (not sure which country you are in)).
If you claim on your private insurance you'll be asked if you have any other insurance, and you have to be honest. If the state is paying for your treatment, you won't have any fees to pay anyway.
Yes there is a general principle in insurance that you cannot claim the full amount under two policies for the same event. You could not, for example, insure your car with two insurers and expect them both to pay you the full value if it is stolen.
If one of the policies pays a flat rate per night just for being in hospital and the other pays the actual cost of treatment that would be legitimate. But you can't claim twice for the same actual cost.