I solved this (error code 51330) by changing my wireless security mode from WEP to nothing (no security) I realise this is a risk but it worked. The next step is to try WPA. Technically WEP security is insecure anyway as people can easily crack into it if they want!
Other's have posted that changing the wireless mode from G to B + G (mixed) works. Others have also reported that switching your wireless router from 11 to 1 or something else works. For me it was switching to no wireless security that did it.
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After a frustrating period trying to get online, what I ended up doing was turning off the security settings on my wireless router (to confirm that I had everything set up properly at that end). Once I got the Wii connected that way, I found that I couldn't get it connected with WPA2, but dropping back to WPA worked OK, and at least is better than WEP. Don't know how people do this without lots of online help...