3G nano won't accept video

I'm having what I think is an unusual, possibly unique, problem with a third-generation iPod nano (8GB) that I purchased yesterday. Everything about the iPod works perfectly except that it won't accept video content. When I select the video content I want to sync to the iPod, I get a message saying that that content is not going to be sync'd because video can't be played on that iPod. Video from my media library can't be dragged onto the iPod icon, either.

I have the latest version of iTunes installed, the latest version of OS X, the latest version of QuickTime, and the latest update of the software for the iPod itself. I might be able to understand this if the video content I'm trying to sync were of my own making (e.g., DVD ripping) and perhaps wasn't encoded properly or something. But this content is directly from the iTunes Music Store, downloaded last night -- a music video and an episode of a TV show. These videos play in iTunes without any problem, but I can't get them into the iPod. I checked to make sure that all purchases were downloaded, and they are. Also, the computer is "authorized."

As I said, this seems like a very odd problem because the iPod is functioning in every other way. And the message I'm getting isn't saying "Your video content is corrupted" or something specific like that -- it's displaying the type of message I'd expect to see if I attached, say, a third-generation iPod and tried to sync video to it. Obviously it would say, "Your iPod can't play video." The third-generation nano obviously can, because there's a video entry in its menus. And when the iPod is connected to the computer, the iPod screen in iTunes offers the option to "Manually sync your music and videos" -- according to the iTunes help file, it would say "Manually sync your music" if it didn't detect a video-capable iPod.

I've been on the phone with Apple tech support and the best they were able to do was assign me a case number and tell me to call them back in a few days.