You are in which version of Exchange 2003? Whether you are on Exchange 2003 Standard Edition and prior to service pack 2, then you have a maximum limit of 16GB on the entire database(saying about physical space). If you are in post SP2, then the limit goes up to 75, but it has to be set to its higher limit. Also you will get this message like Exceeded the Size Limit on Your Mailbox when your data size approaches to its maximum limit.
Whether you are on Exchange 2003 Enterprise Edition, then your limit is set to 8TB. This is a theoretical limit however, it is really based on the amount of space on your disk which you have available up to 8TB. Also, you can use the same registry keys as in the document above to configure lower limits which help not to run out the disk space on the server and to avoid Mailbox size limit exceeded issues.
Also, there are limits set to mailbox and also to the mailbox store as well. And if you clear the whole limits on the mailbox, you should notice that the mail-store has higher enough limits it is because a mailbox with no limit set will only be held to the mailstore’s limits.
It has been experienced that if you have mailboxes larger than 4GB then your mailbox is vulnerable to data corruption and then it is true that you will get an alert box which informs like Exchange Mailbox Exceeded size limit.The process to move or migrate Exchange Mailbox items is extremely a huge head-burn until you moved on Exchange 2010.