Almost the exact thing happened to me the past two weeks. First I had problems getting Apple to send the verification number. I signed in to iCloud.com, and checked all of my settings on both the website and my iMac and finally got that problem to resolve.
So now when I enter username/password, Apple sends a verification number immediately to my iPhone. But when I enter the verification number, Apple TV just kicks me back to the password. Enter password, get verification number ... rinse and repeat ... until blood pressure rises to dangerous levels.
We ended up just watching the movie on my cable provider's on-demand option. Guess what Apple? Same price, and I get to watch the movie immediately without a three-step verification system that doesn't go anywhere. So how do you think I'm going to get my content in the future? And what are the chances that I'll spend $140-$200 on an upgraded Apple TV? Apple needs to get its act together, because this life-long, all-Apple user is not happy at all with the current product/service Apple TV provides.