I have a Rogers Galaxy S3 ... I just did a replacement, after 11 months with the phone -- I was having the exact same problem. I have an after-market charger that charges my phone just fine, but the one that came with it, unless I hold it at EXACTLY the right angle, pray to the right Gods and don't-move-a-muscle ... it will not charge. I cannot use it to connect to my computer at all. I would also THINK I had it at the right angle, leave it over night, wake-up elated at the 100% charged display, only to unplug it and discover 2 minutes later, that it was at 5% or "critical". But I knew it was not a battery issue -- because when I use that one other charger, I can get it to stay in the right angle, and it will hold a charge all day long, with LOTS of use.
Finally I decided it must be a data-port issue. I called Rogers and they agreed to send me a "refurbished" unit to trade it with. So, it arrived 2 days ago. I plugged it in with the cable/charger that came in the new box. I left it overnight. I awoke to exactly enough charge to turn the phone on, but it says "critical" low battery. So I plugged it into my after-market charger -- it charges no problem.
I do NOT believe it is a battery issue. I believe it is a shoddy data-port issue. I believe the size is all wrong for their crappy mini-USB plugs. I am pretty peeved that the replacement is doing exactly what the original was doing. Both hold a charge fine -- so long as I use my own cord and not the one that comes with the device.
I don't even know where I got the other cord .. it may be left over my from Crackberry, but it doesn't have the symbol on it, so I'm doubtful. I wish I could go and purchase 10 more of those chargers, but, like I said, I don't know where it is from. I tried a dollar-store version of the mini-USB, which I never had problems with for the Crackberry. But nope, the g3 didn't even recognize it.
I am stuck using Kies to transfer files to my computer -- I cannot get the Bluetooth to connect on my MacBook and none of the mini-USBs will work any more (the original one, with the original phone, worked fine for nearly 10 months before it started not recognizing the device)
The guy at RadioShack (er, wait, umn, The Source?) said he has an extended battery charger thing attached, so he never has to deal with that. But it makes the phone twice as fat and wont fit my OtterBox (since I drop the phone too often, I can't do without it!)
Good luck .. sorry for the long rant. My point is: it's probably not the battery
(oh, and mine started in around June-ish, I got the phone Sept 30th 2012, so about 9 months in ... and it started out just dropping to 40 or 50% .. and for a while it seemed just fine, so I didn't bother with it, but then it got worse, and my 1yr manufacturers warranty was coming close to an end, so I dealt with it, only to realize it seems like it isn't dealt with at all ..... sorry I don't have better news for you!)
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