That specific error happens when the database server's maximum connections are reached, not when your specific maximum connections are reached. If it was you, you'd see something like "Max Connections" so this means you're not hitting your 10 connection limit.
I've asked our system administrators if they can increase the maximum amount of connections to the database server, they said that it's not being overloaded, so it can handle the amount of connections it currently has.
We constantly monitor our database connections, unfortunately due to a large amount of Joomla and Wordpress attacks lately the connections have been under more pressure. Increasing the amount of connections to relieve the pressure would not help as these connections would all be used up as well at times when the sites are under attack.
We will continue to monitor the situation and disable sites where necessary.