I plugged my iPad 3 in to charge after using it extensively (it had reached around 5% battery remaining) and this morning when I checked it, it had stopped charging at around 85% and was displaying "not charging" by the battery icon.
I unplugged it, replugged it, and although it briefly reported that it was "charging" again, it stopped shortly thereafter.
Although I was using a genuine 10w Apple OEM power brick, I wasn't using an Apple OEM cable. I was using a third party cable because it's slightly longer (70 inches) than the (ridiculously short) Apple docking cable (which is about 39 inches long).
I'm not certain where I purchased it (probably Amazon) but it has the following markings on the side "AWM E164571 2725 VW-1 80 C 30V" and "BizLink Technology." It's relatively heavy in gauge (and thicker than the Apple cable) and it always worked fine with the iPad 2.
I replaced the third-party cable with the Apple OEM cable and it charged the rest of the way, or so I thought, without issue.
Ray Soneira, President of DisplayMate Technologies Corporation, explains what might be happening:
I've now heard from several people regarding this issue. One person told me they traced the problem to their Griffin cable and "tossed it."
The charger itself says 5.1 volts 2.1 amps. That is a fair amount of current. For a short cable thinner wires won't make a difference, but on longer cables it will. Also, the new iPad does draw more power from the AC adapter than the iPad 2.
According to the Apple website the "not charging" icon appears when there is insufficient power from the charging source. This can occur under many circumstances: you are using an iPhone charger or non-Apple charger that has too little power. They might be ok for the iPad 2 but not for the bigger power needs of the new iPad. You can also get this if your computer USB port doesn't have enough power or you are using a non-Apple USB cable (which supplies also supplies the power) that is too thin or too long because of the power losses involved in the cable. Finally, do not use any aftermarket adapter that speeds up charging by exceeding Apple's maximum charging rate of 2.1 amps at 5.1 volts.
You can read more about this issue in Soneira's excellent iPad shootout article, especially the sections on display backlight power consumption and running time on battery.