Sounds like a defective charger. Contact the reseller.
Try this first - Reset the iPad by holding down on the Sleep and Home buttons at the same time for about 10-15 seconds until the Apple Logo appears - ignore the red slider - let go of the buttons. (This is equivalent to rebooting your computer.)
The quickest way (and really the only way) to charge your iPad is with the included 10W or 12W (5W on Mini) USB Power Adapter. iPad will also charge, although more slowly, when attached to a computer with a high-power USB port (many recent Mac computers) or with an iPhone Power Adapter (5W). When attached to a computer via a standard USB port (2.5W, most PCs or older Mac computers) iPad will charge very slowly (but iPad indicates not charging). Make sure your computer is on while charging iPad via USB. If iPad is connected to a computer that’s turned off or is in sleep or standby mode, the iPad battery will continue to drain.
Apple recommends that once a month you let the iPad fully discharge & then recharge to 100%.
How to Calibrate Your Mac, iPhone, or iPad Battery
At this link http://www.tomshardware.com/reviews/galaxy-tab-android-tablet,3014-11.html , tests show that the iPad 2 battery (25 watt-hours) will charge to 90% in 3 hours 1 minute. It will charge to 100% in 4 hours 2 minutes. The new iPad has a larger capacity battery (42 watt-hours), so using the 10W charger will obviously take longer. If you are using your iPad while charging, it will take even longer. It's best to turn your new iPad OFF and charge over night. Also look at The iPad's charging challenge explained http://www.macworld.com/article/1150356/ipadcharging.html
Also, if you have a 3rd generation iPad, look at
Apple: iPad Battery Nothing to Get Charged Up About
Apple Explains New iPad's Continued Charging Beyond 100% Battery Level
New iPad Takes Much Longer to Charge Than iPad 2
Apple Batteries - iPad http://www.apple.com/batteries/ipad.html
iPhone: Hardware troubleshooting (Power/Battery section also applies to iPad)
Extend iPad Battery Life (Look at pjl123 comment)
New iPad Slow to Recharge, Barely Charges During Use
iPad: Charging the battery
Best Practices for iPad Battery Charging
Tips About Charging for New iPad 3
How to Save and Prolong the battery life of your new ipad
Prolong battery lifespan for iPad / iPad 2 / iPad 3: charging tips
iPhone, iPod, Using the iPad Charger
Install and use Battery Doctor HD
To Extend a Device’s Battery Life, Get to Know It Better
iPad Battery Replacement
In rare instances when using the Camera Connection Kit, you may notice that iPad does not charge after using the Camera Connection Kit. Disconnecting and reconnecting the iPad from the charger will resolve this issue.
I've tried resetting it too; no change. And I don't think it's the adapter since I've tried two of them. It would be mighty strange bad luck to receive a faulty new adapter, and also have the 10W adapter I already had -- which works fine with everything else -- suddenly go faulty at the same time.
On a lark, I tried going ahead and leaving it connected to the 10W, even though it claims it's "Not Charging." Apparently it's fibbing, and it is, in fact, charging (albeit slowly). So apparently the problem isn't that it's not charging with the adapter, but rather that it falsely says it's not charging with the adapter. That -- though odd -- I can live with.
It does sound like an obscure hardware issue with the iPad. When you connect an iOS device to a charger, there is some identification that goes on between device and charger. Although the iPad, iPhone, and Touch have the same dock connector, they each have a different ID resistance on one of pins (not sure if this exists with the Lightning connector). I think this is used to set the charging rate.
The fact that your iPad charges with a 5W charger is nothing novel since they all will charge with 5W (very slowly), but what is interesting is that you don't get the same error that you do with a 10W charger. That tells me that the sense resistor inside your iPad may not be connected correctly to identify it as an iPad, so the iPad thinks the charger is not charging but it really is. I don't know that for a fact, but it does seem like a plausible explanation. I'd take it back to the reseller to show them, and have them show you that it is OK if they insist.
Texas Mac Man: Yes, both of the adapters I've tried and gotten "Not Charging" messages with are 10W USB Power Adapters.
rockmyplimsoul: I like your theory. Even if it's not true, it seems plausible. Unfortunately, I can't just take it back to the reseller to show them. I ordered it from an Amazon Marketplace reseller, so taking it back would mean removing it from its OtterBox, mailing it back to them, etc. It's not a big enough problem to hassle with that. If it turns out that something needs to be repaired/replaced, I'd more likely just take it into my local Apple Store to have it fixed.