Connection limits reached.
FOR FTP
Most clients open several simultaneous connections. Reduce the number in the client settings.
FileZilla: Settings > Transfers > Maximum simultaneous transfers. Two or three is usually sufficient.
If you have several FTP sessions open, close them.
Connections also linger after disconnection. Waiting a few minutes usually resolves it.
FOR MAIL
A client opening many connections while synchronising large mailboxes.
Reduce the check frequency and the number of simultaneous connections in the client's advanced settings.
Several devices synchronising the same mailbox simultaneously compounds this.
FOR THE DATABASE
Covered separately. Usually an application leaking connections or several sites competing.
FOR CPANEL
Multiple simultaneous sessions. Log out properly rather than closing the tab.
THE GENERAL PATTERN
Something is opening more connections than it needs, or not closing them.
Reduce concurrency in the client, and wait for existing connections to time out.
