I've had this happen to me twice. Run out of battery and then charging won't work.
The suggestions: "check the connector" don't help: you can verify the connection is good because the phone detects the charger and shows "sign-of-life" when you plug it in. Users like me then assume things are ok and walk away.
What seems to happen is that the phone charges for about half a minute and then decides it is time to turn on. Apparently the phone wants to be on during charging. Fine. However, somewhere during boot, it turns off the charger(*), planning to turn it back on later on during the boot. But it crashes/turns off due to low battery before it gets to the point where the charger(*) is turned back on.
I contacted motorola support about this. They have a "you can try" this. suggestion.
After having been plugged in for a while (IMHO after the try-to-boot nothing happens, the current draw from the powersupply is zero), press and hold the power button for two minutes. Then let go and you should get a boot menu. Try the normal boot or the recovery.
Back then I didn't know for sure that "recovery" wouldn't wipe my phone, so I only tried the "normal boot". As was to be expected it did exactly the same as during the "normal boot", i.e. turn off half way through the boot. Or flash the notification led three times ("not enough power to boot") and then shut down.
Anyway, both times this happened to me my colleague said: Hand it to me, let me try. And after a minute or two the phone starts charging and we decide: "don't touch it", and then it works again until it ends up at 0% again....
So when this happens to you... just hand your phone to my colleague, and he'll have it charging again in no time... You don't think this is a good solution? No? Neither do I.
Motorola further adds: If that doesn't solve it, send it back for repair.
My opinion on this matter is that it should simply charge if you plug it in when it's dead. It is "not perfect, but might be acceptable" if there is a "guaranteed to work" procedure that gets it to charge again.
The situation now is that after some fiddling with the phone it sometimes starts to charge again.
I'm considering "letting it run down again", then trying neofreak's suggestion above that I haven't tried, and if that doesn't work, send it in for "repair". This can be planned for some time when I don't need the phone (that much).
I have to plan this when I can switch phones back to my old phone, as I'm not convinced that my colleague will be able to "perform" again next time. He might have been lucky fiddling with the phone until it randomly started working again.
Oh, one thing I'm going to do next time: Motorola says to use an AC charger. The thing is you probably have to use an AC charger that is detected as such! Some of my AC chargers are detected as "USB". Settings->battery then you can see how your charger is detected.
For the technically inclined: when the datalines (USB_D+ and USB_D-) are connected together the phone detects this as "ac charger".
The motorola charger allows charging at around 900mA (at 5v), while the above D+/D- short only results in 500mA charge current. I would still like to figure out how to get it to charge at > 500mA, as I have a 30A@5V powersupply on my desk that is able to handle anything the phone might want.....
(some modern RC-flying batteries are able to charge in about 12 minutes at 5 times their rated capacity ("5C"). So that would mean charging the 'G at 6A during 12 minutes. Sounds neat. Alas this battery, the phone and the cables/connectors are all probably not up to that job.... Anyway, my power supply would be able to handle that no sweat!)
(*) Here "charger" refers to the chip inside the phone that does the "5V-from-micro-USB -> LIPO battery" function...