Bloodborne does support the newly launched Suspend/Resume feature of PS4 firmware 2.50, but only when you're playing Offline Mode.
Checking out the new feature this morning VideoGamer discovered that those who choose the Online Mode - which allows other players to enter your game - will be chucked back to the game's menu when attempting to resume after suspending.
It you want to make use of the handy resume function for Bloodborne you'll need to choose Offline play.
It's hardly surprising given the data connection will be lost once you suspend the game, so there's no option but to kick you out and establish a new connection.
Bloodborne launches in the UK tomorrow.