i hope at least some of you might come back and read this:
I have the beginnings of the same problem on a v1.0 xbox, modded with a solderless pogopins xbit chip.
things just magically started to get worse over time. i tried bios and other software upgrades, nothing helped. about this time we noticed that the 3 and out thing happened mostly after watching a dvd, which means after booting into the msdash. also, it seemed to happen more often after xbmc timed out and automatically shut off the xbox.
so we didn't do that.
but then the 3 and out thing started happening again, but i could get the xbox to finally boot after smacking it a couple times. one day, the smacking didn't help enough, so i took it apart a reseated the mod chip after running an eraser over all the electrical contact points.
this seemed to work very well for about 2 months, and then yesterday i tried a dvd again, booted into the msdash, and then part way through the dvd the box seemed to hang a bit, but didn't immediately frao or frag.
however, when i tried to reboot, i got the 3 and out. i was able to smack it into working last night, but this morning no amount of external abuse would get it to boot.
SO, this morning i tired reseating/cleaning the points again, and it seems to be working again. enough so that i've backed up (on pc) the hd lock code, and i plan on unlocking the drive later (i installed a 300GB 5400rpm drive a while ago) to facilitate moving it if the 3 and out persists.
ANYWAY, the above is just a laundry list of my symptoms. so, lets draw some conclusions:
From what you guys have posted, and from my own experience, it looks like we are looking at a hardware issue. Since mine seemed to be influnced by the dvd spinning up (i never use it otherwise) OR by something triggering a timed soft-off, my thinking is (similar to you) that there is some dodgy device that is usually ok, but when it sees a large power draw, it switches to a bad mode that is very hard to get it out of. now, obviously, mine was also related to some mechanical issue since smacking the box brought it back to life. this could point to an issue with the pogopins or maybe with the main power connection. i doubt any other solder joints would be effected by a smack.
if there is a capacitor that has gone bad that needs to stay charged for the board to work right, then there is a chance that a large power draw (or a shock like a soft off) might discharge it and it would have a hard time recharging (if at all), a problem possibly influenced by a poor D0 contact, or some other bad contact with a mod chip. if the capacitor is there to smooth out the 3.3v you guys are missing, then it could be going bad and acting like a short, dropping the voltage down. also, whatever regulator is being used to generate that 3.3v might be working really hard to supply that 3.3v even against a leaky cap, and so it's usually on the edge and gets thrown off it when the system gets shocked.
so, wild guess, there is a leaky cap paired with an overworked regulator. a fix might require replacing both. testing the cap might be hard, because it might not be bad except under load.
OR, for some of you who have giveaway boards, you could try wiring a known source for 3.3v from another place on the board (there are probably a couple) to where pimplex is seeing a lack of the right voltage. this might fry the board, or it might bandaid the problem for a short while... i suppose if the problem is not a leaky cap and is instead soley a bad regulator, then this could be a semi permanent fix, but my guess is that it'd be short term only. you never know, though.
if my problem persists and/or gets worse, i'll put more time into trying to nail a fix down. till then, it is my hope that some of you with some time and already broken boards will try again...
one last thing, someone above said they tried replacing the cap that net(something) (don't remember the name) said might be the problem (the timer "battery" cap), but replacing it seemed to cause more problems... that cap most likely has a polarity, and probably doesn't like heat very much. if you put it on backwards or weren't really careful about keeping it cool, you could have easily turned it into a short... i'd love to see a pic.