Will display a red warning in all pages of the suite if the member has the Inbox full. This is included in Kitchen Sink with an additional option to warn via a top of page bar instead of included block.
If you want to dismiss the message for a specific user group, just enable the Override recipients' storage quota? setting for the group.
The message itself is a language bit, so if you want to change it, go to ACP -> Customizations -> your language -> Translate and search for lang key full_box_msg and change its content to whatever you want.