I was installing battlefield 4 and it had gotten about half way, I decided to look at the Playstation store but it wouldn't load because psn is having issues and by the time it took me to button out of the store the game was done installing. I'm sure when I watched the ps4 live stream it took battlefield a while to fully install?
It took me a really really long time to completely download Battlefield 4, had to basically keep it running all night so I don't know. NBA 2k14 installed in about the time it took me to play one NBA game but that was from the disc but is also a 50 Gig download.
You only installed enough of the game as was needed to make it playable. There was still more to install, leaving certain aspects of the game inaccessible. For example, I believe Battlefield automatically downloads and installs the multiplayer portion of the game first, which means you probably could not play the campaign the first time you booted up the game. The game then continues downloading/installing as you play it.