When powered on, my One would charge using pretty much any standard USB charging cable and wall charger, however, if the battery drained all the way and the phone powered down, it would ONLY work with a cheapo Chinese wall plug and power-only micro USB cable. Weird, right? If I hadn't had this random cable lying around (I can't remember what it's from,) my phone would have been effectively bricked.
SO....after searching the web to no avail, I finally called Verizon who did say it was covered by warranty, but they wanted to check with HTC first, which is where I learned this little trick:
To recalibrate your battery to a specific charger (or in general) do the following:
1) With phone on and plugged in to the charger (kinda a problem if your phone won't charge...I suppose you could try it with it off as well...) Hold ALL the physical buttons down for 3 minutes (Power+Vol up/down) The phone will cycle a bunch of times (you'll see only the HTC logo.)
2) After the 3 minutes is up, leave the phone plugged in and don't mess with it for 10 minutes (I'd say leave it whether it is on or off at the end of the 3 minutes, if it's off turn it back on and let it sit for another 10 minutes or so, that's what I did.)
That's it! Apparently this calibrates the battery to the charger you are using & it did the trick for me. I can now let my phone die & plug it in using any charger I have & it will actually charge. I'm not sure if this will help with any other battery problems people may have, but it might be worth a try if you've exhausted your alternatives.