It doesn't allow you to receive any more messages and people see a message when they try to send you a PM saying that your inbox is full.
At the bottom of screen for the Inbox and Sent Items (and any other folders you created), there is a link for you to download all of your messages in either an XML, CSV, or Text format. Download all of the messages in each folder and then delete them to free up space.
Are you running out of space?