My Google Apps Outlook Sync Tool is telling me my "Local mailbox is full"

This is common, and simply indicates that the local Microsoft Outlook copy of your Google Apps for Business email is limited to a certain amount - by default this is 1GB.  It does not mean that your account will stop receiving email.  This only affects how much email is downloaded/synchronized with Outlook on your computer.  You can change this setting to "Unlimited" or several other data amounts by following the directions here: 

http://mail.google.com/support/bin/answer.py?answer=184126