Persistent "Your database has already been upgraded to CiviCRM" Problem
Huge thanks to @PradeepNayak for your help! The problem wasn't with the paths in civicrm.settings.php, but that suggestion led us to look at our configuration. Turns out that we had some confusion between a development and staging server, so were, in fact, not updating the right instance. In essence, the version number from the database was right, but it was looking in the wrong place for the files version.