New Ipad (3) won't pair with devices

Question

I have two separate speakers that I used with no problem with my iPad 1. I replaced it with the New iPad and neither speaker now works. Do I have a defective iPad that needs replacing?
Details: the New iPad is 16G, 4G-capable but I choose to use wifi only and have not bought a plan. One of the speakers is the Bose Soundlink. I forget the brand of the other one. Neither has ever successfully paired. The Bluetooth symbol does not appear on the upper right-hand corner of the iPad screen most of the time. (It does occasionally when I fiddle with turning Bluetooth on and off through "settings"). When it "searches" for devices, the only device it ever finds is my useless Blackberry. Neither of the 2 speakers even show up. I've tried resetting the devices, and resetting the iPad including a complete reset that involved restoring it from an iCloud backup. Nothing works.

Answer 1

Not entirely accurate.  Per a kb article at Apple.com:
Apple peripheral devices may request more than 500 mA (Milliamps) at 5 V (Volts) from a port to function or to allow for faster charging. Such Apple peripheral devices include:
Apple MacBook Air SuperDrive (when connected to supported computers)
Aluminum Wired Keyboard*
iPod
iPhone
iPad
To meet requests for additional power from these Apple peripherals and devices, some Apple computers and displays can provide up to 1100 mA at 5 V through the port to which the Apple peripheral or device is connected.
For whatever reason, my iPad has been much better for most of yesterday and today.  As others have noted, it still does not increase it's charge while in use (nor does it discharge much, as though the charger is providing just enough power to run the iPad while in use).  I think when I was first restoring everything and loading up apps, etc., it was running VERY HOT which probably meant it was drawing a lot of power from the battery, and the charger could not keep up (which meant the battery was slowly discharging). 
Now that activity on my iPad is more normalized, it's not as hot, but still doesn't charge much while in use.
I'm sure Apple can do something to increase the charge rate so it does charge faster when asleep.  My iPad 2 charged a bit too slow for my tastes and now the new iPad is even worse!