I had about the same problem for about an 30 minutes today. I use the DS Dongle to connect my Wii to the internet though, and sometimes it has trouble creating an IP, although today wasn't the case. I got into selecting Mastercard/Visa screen to get more WiiPoints for Contra 3 and Mario Kart 64, and then all of a sudden I couldn't connect after that. I tried a few times and wouldn't even let me reconnect to the VC Channel area. After about 30 minutes of thinking it was my dongle and getting it to work, I have a feeling it might have just been some type of downtime for Nintendo. I finally got it to work and downloaded the 2 games, but I definately was having a problem connecting earlier.
Edit: I don't think it was my dongle because I used it to connect through Wi-Fi on my DS, and I was connected to put my Credit Card info, then it disconnected me before I could process my credit card.