I've never seen a bin itself break, but I've had several times where a latch was broken when a passenger tried to bang it shut on a too-big bag. Believe it or not, we can't take everything out of the bin and just leave it open for the flight when the latch breaks. We have to call in maintenance, they have to tape the bin closed and it has to be written up in the aircraft's logbook. It seems silly considering that anyone with a roll of tape could perform this particular bit of "maintenance". Still, that's the reg and I've had a few flights significantly delayed waiting for maintenance to come out and MEL a bin.

So, while the FA's announcement was probably overkill, there was a bit of truth at the bottom of it.