DB error: No such field (Civi 4.7.15)
The solution was to upgrade the database following my CiviCRM version update. Detailed instructions for doing this are at https://wiki.civicrm.org/confluence/display/CRMDOC/Upgrading+CiviCRM+for+Wordpress#UpgradingCiviCRMforWordPress-4.RuntheUpgradescript

Once I had completed this relatively simple process, I was finally able to add payment processors without throwing errors. And just as exciting, the System Status in the Civi admin footer finally switched from "Critical" to "OK."