Basically if you connect to a LIVE game server your download will stop. This can be playing online, leaderboards, etc. If you want, just try it out and see. You can always restart the download later.
Also, you can enable background downloading. This will allow you to turn your console off, but it will remain on enough to download your game. When it is done, the console will turn itself off. You can access this by navigation to System Settings > Console
Settings > Startup and Shutdown > Background Downloads.
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