Well, this issue didn't repeat itself. It may have been a non-PCo-related fluke, or it may be that my actions described below resolved the problem.
Steve, your answer reminded me of your answer to my previous question "With numerous PCo agent instances, agents don't load on server restart". Therefore I did the following, which I'd recommend for all PCo users. Coincidentally or not, after I did this, the problem didn't happen again.
1. Make sure that each Agent Instance has a startup dependency to Windows service it is reading from
2. Using the Windows Services control panel, set the startup mode for each Agent Instance to Automatic - Delayed Start.
3. Make sure that the Windows service that PCo is reading from is set to start Automatically (not delayed start).
