Well, now I don't think it was Google Sync after all. I installed a few CABs, rebooted, installed a few more, rebooted, etc. Three times so far, when I rebooted Bluetooth didn't load. Only once have I been able to uninstall what I installed between reboots and have Bluetooth start. I've had to hard reset twice and start all over again.
The CABs I successfully uninstalled were battery management CABs, but I'm still not convinced that they were the cause. I'm leaning toward a hardware problem, which other people have had, but it's not consistent.
Does anyone have any other suggestions of how to narrow down the problem?