If you fail a case (by getting killed, knocked out, or by being spotted while tailing someone, etc.), you will be given an option to restart the case and it will put you back at the most recent save point in the case.
If you mean charging the wrong suspect, usually you will still complete the case, just with a lower star rating at the end. In this situation, you can still continue the game. The story is pretty linear, so if it is necessary for the story for a certain person to be charged, then there won't be an option to not charge them.
For more information on successfully completing cases and getting five star ratings, follow these LA Noire Case Guides.
