My iPad has a Bad battery-is that possible?

It certainly is possible. Since your symptoms do match those of a defective battery, you should probably take it into an Apple Store to get it replaced for free (if it's under warranty), otherwise it'll cost some money, but not as much as a new iPad. It may just be a software bug, but the Apple Store should be able to help you.

Also, battery issues can be tested automatically online at:

• https://www.apple.com/support/ipad/repair/

Choose battery and enter your contact information and you will get emailed/messaged a test diagnostic. Once the diagnostic is run, you can chat/talk directly with Apple to see if it's something they can repair and whether you'll pay for the repair or if Apple will pay for it.