If the same thing happens with 2 different chargers with 2 different cables, the issue is at the iPad.
Is the iPad recognized when you connect it to iTunes on a computer? I recommend you try it.
1. Try doing a hard reset:
Press and hold both the Home button and the Sleep/wake button simultaneously for about 15 seconds and release when the Apple logo appears.
2. If that doesn't solve the issue, backup the device and restore it as new.
In that case, use: http://support.apple.com/kb/HT4137
NB: set up as a NEW device because a software issue will be in the backup. If the issue persists after restoring as new, you should offer the device for service.
If you get specific error messages/codes in iTunes when updating/restoring, check this: http://support.apple.com/kb/ts3694
If you're not able to restore the device, this one might help: http://support.apple.com/kb/HT1808
Using that last article will erase everything on the device, so make sure you have a backup!
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.)
And this - Unplug the charger from the iPad and then replug. May resume charging.
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
Extend iPad Battery Life (Look at pjl123 comment)
New iPad Slow to Recharge, Barely Charges During Use
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.
Any ideas on what could be wrong? I suspect the port but I'm hoping it will be something a lot simpler and cheaper to fix.
The port, the cable, the battery .... Its really hard to tell unless you test with known good accessories and apparently you have tried that, so from here its anybody's guess.
Close all apps and Reboot the iPad and see if that makes any difference at all. Rebooting resets the system and that might help, but more than likely, your next step would be to restore the iOS software if you have exhausted all possible accessories issues.
From your home screen - Double tap the home button and the recents tray will appear with all of your recent apps displayed at the bottom. Tap and hold down on any app icon until it begins to wiggle. Tap the minus sign in the upper left corner of the app that you want to close. Tap the home button or anywhere above the task bar.
Reboot 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 if it appears on the screen - let go of the buttons. Let the iPad start up.