In case anyone is curious, I identified two sources of this issue.
1) JGit is throwing exceptions because nanosecond resolution is set to UNDEFINED and any time you do math (add a value) with that value in Java long, you get an Arithmetic Overflow. This seems to have left some threads in an indeterminate state.
2) More importantly, we had an out of date SSL certificate. There were no exceptions thrown during startup relating to this, but we could not get to the update site or download metadata on startup. I think this was the root of many problems but it would have been nice to get an exception (and one with the URL involved, which Jenkins plugins are notorious for not reporting).
Cheers,
Randall