My associate suggested I check the web.config file and see that the authentication credentials were set correctly for connecting to the database. He thought that maybe this MS SQL Server 2005 Express had authentication set to be either Windows or SQL (Properties > Security - in Mangement Studio) and that one set of credentials were not authenticating correctly.

Turns out SQL Server 2005 Express was set for Windows Authentication only and those credentials were set correctly. But when I got to thinking about credentials, I found an old MSDE instance running for another database (Plesk SiteBuilder) that had SQL Server authentication only enabled. Since the database managed by that MSDE is no longer used, I disabled the MSDE instance altogether. Since then, about two weeks ago, the error DNN Upgrade Error that appeared every couple days when connecting to a DNN portal is gone and has not occurred again.

Maybe the MSDE instance (w/ SQL Server authentication) was somehow causing trouble with the MS SQL Server 2005 Express engine handling the DotNetNuke databases. Thanks to my associate Sebastian for lighting the way to a solution.