That's an error from the MySQL server due to the max allowed concurrent connections being set too low for your site to properly work. You would have to increase the value of max_connections in the MySQL server config my.cnf. You might also want to set a value for wait_timeout if not already set or decrease the value if set. The wait_timeout value is the time in seconds the server waits before closing an unused connection. Default is 28800 seconds (8 hours) which means that it might not be ablue to open a new connection (due to the limit of 15 set for max_connections) even if there are less users on your site. Setting wait_timeout = 600 works for me.

Increasing max_connections will result in more memory being used by the MySQL server. So, be careful with increasing.

If you are on a shared server with no permission to alter my.cnf of the MySQL server, you would have to get in contact with the support to ask for help. If they don't allow more than 15 connections you might need to change your hosting plan or hoster.