I've had exactly the same problem...
I was using my phone and battery was going down. I plugged it in and it just said on screen "Not Charging!" so I unplugged and plugged back in. Still same thing. Again and again.
So anyway, I left it plugged in and coz I was in a call, the battery ran out and I obviously got cut off. Anyway, when I tried to turn it back on I got to the pin code bit, then the security code bit... then it just turned off again.
I unplugged the charger, plugged in back in, and in a few seconds then phone came up with the nokia logo, lit up and then made a decending beeping sound like the sound it makes when its running out of battery.
Then the battery icon comes up but the screen then turns off and... here's the really wierd bit... it carries on doing this in a loop for as long as the charger is turned on! Logo comes up, icon comes up, makes sound, turns off again and again!
I have tried a different battery direct from vodafone and its all the same.
I got so annoyed I almost threw it against the wall! I've got 3.12 not 3.16 so it may not be the software version.
OK, so other things I have tried...
1: I can actually get the phone to turn on for a few seconds like this:
Hold down the power button with the charger disconnected.
As soon as the screen lights up, plug in the charger. (if I dont plug it in, the screen turns off again).
This seems to sustain the phones dead battery but only until you enter the pin and security code, then the phone goes dead again, probably because the phone uses more power to log onto the network.
2: Without the sim card, the phone will still do the same loop thing but when you use the method above to turn on the phone, it just says insert sim card. (Dur) lol.
3: I have actually had this problem before AND got the phone to work again, but I was drunk when I did it and have no recollection as to how I managed it. But I do remember turning the phone on and taking the battery out and putting it in again afterward... hehe. Coz I cant remember exactly it's hard to say.
Oh well, if anyone else has any insight into this little prob i'm all ears. By the way, I do have it on standard warranty as I've only had it 3 weeks. Will get it replaced by vodafone soon if I cant get it to work at all.
If I manage to make it work, I'll be sure to update this thread. Anyone else had this problem?
Rob