Thanks, Karl, for taking up this issue!
So far, I've only seen it with the two applications, InDesign and InCopy. The error message only appears the first time one of the applications is launched after a reboot; subsequent launches of the applications during the same logon session do not trigger the error message, and logging out and logging into the same account doesn't trigger the error, either. Interestingly, the error message isn't the top-most window; it's displayed behind the application windows, such as the Welcome Center in InDesign. It might actually be the bottom-most window in the window hierarchy, actually. Can't 100% confirm that, though.
If I click the Quit button in the error message, the application does NOT quit. InDesign continues to work normally, as best as I can tell.
Thanks again Karl!