Make sure that you are attempting to access the section of the game you have started to download. For example if you have started downloading online multiplayer, make sure you are not attempting to access one-player mode.
Check that the section of the game you are trying to start has fully downloaded. Go to the game either on the content launcher or in [Library], highlight it and press OPTIONS (do not select Start). Choose [Information] from the menu on the left and check Download Status. This will show you how much of the selected section of the game has downloaded. You will notice that the size of the downloading file is smaller than the total size of the game. You cannot start the game until these two numbers are the same.