ORA-12516: TNS:listener could not find available handler with matching protocol

Today my clients started to give me back this error: ORA-12516: TNS:listener could not find available handler with matching protocol.
This is an example of poor error message, in my opinion, and the reason behind the appearing of the above error was, in my case, the xceeding of the available connections
And why do I believe this an example of poor error message? Well, because it does not indicate what the actual problem it is. For example, PostgreSQL provides a clearer message error for a similar situation: psql: error: could not connect to server: FATAL: too many connections for role "luca".