I'm having the same problem. Using Iphone 4, and after the update IOS 7.0.x and now i'm using IOS 7.1 my battery drop dead can't charging and stuck @ Low battery icon.
When My Iphone still ON, charging till 100% (when i a sleep, using Plug Wall Adapter). but the morning iPhone is OFF and there's info Low Battery and can't Turn IT on. After a while try to turn it on again, and finally it can after a short time connecting USB to PC and try to turn ON again, The weird part Battery already 100%, so the info is wrong. For weeks I don't charge till 100% afraid the same problem occured (already 3 times happened when charging till 100%). But this day i forgot, and sleep when my iPhone is charging (iPhone is ON when charging). Now my iPhone drop dead again, still can't turn back on again. And this happen on IOS 7, when charging its no problem if my iPhone ON and not let 100% chargin', and my battery usage is fine so its not a deffective battery.
When the low battery icon is shown, it detect when i'm pluggin the power, but it can't charge even tried 2 hours (before already 100% chargin). Still trying to get my phone turn ON, but i'm sure when it turn ON it will shown 100% Battery Life.. This is s*cks, hope apple will get this fixed soon..
Note:
* i haven't tried to restore to default new iPhone
* My iPhone still responsive when i tried to DFU mode (My PC still detect a DFU Mode)
* Tries after DFU mode and restart iPhone still won't make my iPhone Turn ON (press Power and Home Button for more than 10 second, still don't see apple logo)
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Finally found a Apple Logo (My iPhone back On again), of course 100% Battery stated in my iPhone.
In my case to turn back on again is:
1. Press Power Button and Home Button (hold it). Make sure you usb cable already plug to your iPhone and don't plug cable to any power or PC usb yet.
2. Plug the Cable to PC (still holding the Power and Home Button), if you see Apple Logo you can release Power and Home Button.
3. You iPhone is turn ON again..