Hi John,
Thanks for your test and report. From a thread in Office 2011 for Mac forum, they have discussed this issue may be related to the Apple user account. And I believe you have checked this thread. For such client side issue, re-installing the application is also a good option. I suggest you following the steps in the article below to completely remove Office for Mac 2011 and install it again.
http://support.microsoft.com/kb/2398768
To get the issue resolved, we need the Support Engineer from Office 2011 for Mac Support forum to further troubleshooting. Because we have limited material on the Office 2011 for Mac support. I truly suggest you post the question and the test results you have got in our Office 2011 for Mac forum. The engineers there are more professional with the client side issue troubleshoot. You will get more information from there. You may reach them by the following link.
http://answers.microsoft.com/en-us/mac/forum/macoffice2011?tab=Threads
Here are some similar threads may help you.
http://answers.microsoft.com/en-us/search/search?SearchTerm=document+connection+keep+asking+password&CurrentScope.ForumName=mac&CurrentScope.Filter=macoffice2011&ContentTypeScope=&x=0&y=0
Thanks,
August