The answer to this question varies depending on what state you are in, at least in as far as you are contemplating making a claim against a local or state government. In many instances, there is a shorter timeframe within which to make the claim and it must be done on a certain form and directed to a certain state official. The best thing you can do is to consult with an attorney in your area who has experience making a claim against a local or state government.
The federal government also has very specific claim procedures, and these claim procedures may depend in part upon your relationship to the federal government. Are you in the military? Are you employed by the government? Are you a citizen who happened to be injured by a federal government employee? There is no one clear answer and each case is different.
Again, the best thing you can do is to speak to an attorney in your local area who is experienced with claims against the federal government.