---

Nope. Just gives you the big alert that you cannot miss when you log in.

This panel checks the numebr of messages in your inbox against the maximum number possible. Does it equal or exceed the maximum number, the panel will become visible.
Sending the email would need to be implemented on a completely different level: when a new message is sent to a user, it will need to check if this message will make the maximum number be reached and if it does it need to send the email. That requires modification of the core and cannot be established easily with a panel like this.