Nintendo's chief explanation for Error Code 85010 goes something like: "Ours [sic] servers may be trying to handle an unusually high amount of traffic (especially if you are trying to play online with a recently-released or popular game, or if there is a new Wii system update). Please wait a few hours and try the connection again later." That's an oblique way of stating, simply: You're trying to play Smash Bros., aren't you?
Brawl's online component isn't entirely dysfunctional. If you've traded digits, playing 'with friends' is stable, and also permits more than one player from the same Wii to participate. Likewise, Spectator Mode is almost always working if you're the faceless voyeur-type who also likes to spend a little coin (in this case, on arbitrary betting). But when it comes to random brawls 'with anyone,' Nintendo's Wi-Fi Connection tends to get, um, stressed. Good luck getting anything more than a few rounds with the sandbag before the service drops you during peak hours -- also known as, 'nights and weekends.'
When contacted, Nintendo assured us that its technicians are looking into the problem. See what happens when you hire a pair of plumbers to build your server farm out of warp pipes?
[Note: If you've received Error Code 85010 and are unable to establish any kind of online connection, then the problem could be associated with your internet service (dude, dial-up is so not gonna work -- ever) or your router (Wii can be picky). Refer to Nintendo Customer Service for more information.]