Sorry to dig up an old thread, but it matches the problem my Desire developed today.
It's unbranded and has never been rooted, only ever been run with official HTC updates, and was a solid phone until I installed HTC's 2.3 Desire update and then it suffered infrequent, random reboots.
Recently I've been having problems activating mobile data. The mobile data radio would turn on, but there wouldn't be any activity. A reboot would fix it, but today I rebooted it and now it's stuck in a reboot loop, which has never happened before. I've tried removing the SIM, SD Card, and leaving the battery out for hours all to no avail. It gets to the HTC screen where it writes "quietly brilliant" and then reboots. Repeat ad infinitum.
Apart from the 2.3 random reboots, it has been a solid phone since I got it nearly two years ago. It has never suffered the overheating issues and hadn't been used for a few hours and was cold when I restarted it.
Because it has got HTC's 2.3 update on it, it no longer has a bootloader you can access by holding down a button on start-up, so I have no way of doing a factory reset to see if that fixes it.
Does anybody have any ideas on what I could try?
Edit: I've just discovered holding the Volume - button and then powering on does grant access to the menu that allows me to "clear storage," so I will backup my SD Card data tomorrow before doing a hard reset.