If its not the traffic, then you may be typing it wrong. On the registration card, it has the numbers in all caps with daashes. try entering the code w/o the spaces or dashes and in lower case. That works for me.
It could be either one. But I would wait untill Nintendo's servers get more relaxed. The high volume is due to the great amouts of wii users connecting online.
i tried registering again the other after i created a new account since i forgot my username and password from my previous account with nintendo and i had problem. Probably servers problems.