Chat gets "stuck" or "full", no new messages can be posted.

I've had a problem on a website I've built for my kids where the chat seems to get stuck. You can access the chat but posts sent to it get lost. I can create a new empty chat which seems to work for a while, but eventually also dies. The old rooms continue to be stuck even if the new rooms are working. These are rooms being actively used for weeks on end.

I don't know where to start looking to debug this... The last message doesn't seem like it broke the system, and it worked once for months, then recently only for weeks. Where should I start looking to figure out what's going wrong?