There doesn't appear to be much documentation of this on the net, considering it's a dead easy way of killing your Wii enough for the average owner to send the entire unit back to Nintendo.
Imagine my surprise when switching on my Wii, after not having used it for about a week, to find that it simply would not power up. It's incredibly frustrating - it's not a PC, there's nothing really to fiddle with, it's either working or not, right? I've had consoles since the original NES, all of which, when a problem occurred, would just flash strange colours or what have you. The point is, the power was on so it didn't feel so impossible.
So, after a bit of trial and error I had it figured out. What had happened was we had a black out during the week causing all the power to drop out of the house suddenly. The power supply on the Wii apparently has a 3amp fuse that is very easy to blow provided that the power is not turned off correctly. Now, luckily, mine hadn't blown per se, because I managed to get it working by thumping it (just goes to show you once again that everything can be fixed by a good hit), but from speaking to some other people I found this occurrence is quite common.
If this happens to you, one of your range of nut or hex drivers and a simple 3 amp fuse is all you need to avoid sending the power supply off for an extended amount of time that you could be playing Metroid 3 in instead. Am I right or am I right? Or, if you're lucky, a simple thump.
This shouldn't affect Wii's that aren't constantly powered using the Connect24 settings.