This is strictly related to the Xbox One as I do not know how the process works for PS4.
During the installation process of a game, the notification comes up "Ready to Start".
I've noticed on some games that even though it said it was ready, I couldn't play the game. It would bring me to the start screen but when I tried to start, it would tell me to wait until installation is complete.
Is there any risk to start playing games while they are still installing? Possible data corruption of the install, or save game data?