Why might an Apple TV stop playing purchased content from iTunes?

Question

I am trying to watch television shows that I just purchased through the iTunes store on my Apple TV. Every time I try to play any purchased shows on my Apple TV, it gives me the error message

Authorization is required. To play this content on your Apple TV, choose Authorize This Computer from the Store menu in iTunes on your computer, then enter the Apple ID and password used for the purchase.

I have done this. I have restarted the Apple TV. I have deauthorized and reauthorized my computer. It still gives me this error message. I have tried signing out of and back into the Apple TV, and that doesn't work either. This is the first time I've seen this message in the year + I have had this Apple TV. Also, it was playing purchased content just fine last week. Something has apparently changed since then, and I can't figure out what.

How do I get my Apple TV to recognize this purchased content again?