[Not sure why I missed this ping, but better late than never?]
There is a limit of new private message threads in a day: 20. There’s no limit for message replies, however. (And there’s no way to increase or lift that limit just for senior members right now.)
The upshot is that if you continue a message thread, your won’t be rate limited. But if you start 20 message threads in a day, you’ll see this message on 21. As @skieurope suggests, this is prevent spam and other unwelcome bulk messages.
This is a rolling limit. So it’s not exactly a calendar day, which makes looking for when the limit was reached a little harder. The limit has been hit about a dozen times since we moved to this platform in Dec. 2020. One of those times was a new user who sent “Help!” to 20 users before the limit kicked in. 
We can up the limit if it’s really an issue, but I think the default is fairly reasonable most of the time.