According to this Playstation Forum thread, "Error CE-34878-0" has been causing severe issues. As in, save file corruption to nearly every available title.
PS community members are claiming save file loss in NBA 2K14, BF4, Ghosts, Black Flag, and others. It doesn't seem to affect any specific game nor arrive at any specific time. It just happens, leaving you heartbroken and alone.
Though let's be honest, you were most likely alone before this error anyway.
The worst case of "CE-34878-0" is seen when it renders the PS4 useless, not allowing you to execute any application. Hopefully a fix is in the works, but as of right now there's been no word from Sony.