So similar to ignition's problem, I had a strange problem with my Wii U last night.
I went to hook-up my Wii U in our bedroom, which has an old Sony CRT-TV in there, and I used the original Wii A/V cables to hook it up. And no picture. I used the same set-up before on another CRT-TV I have, and it worked fine, but for whatever reason, not on this TV. I have no idea why, because I just was using my PS2 on the very same TV. I double-checked all the connections, and tried both Video Input and Channel 3 & 4.
So I ended up settling for my original Wii, which worked fine using the same exact cord. The whole reason I wanted it in there was Netflix, Hulu, and YouTube; so the original Wii worked fine for that.