All companies experience this drought to some degree, but looking at GameCube vs. Xbox, for example, I think the Xbox had a few things going for it during the drought period that GameCube did not:
1. Exclusive games (Wreckless and Bloodwake). They may not have been the best games, but Microsoft still ran with them, advertising both like heck. Meanwhile, all Nintendo had was that basketball game, which was half-heartedly advertised alongside Rogue Leader for like the whole first quarter of 2002. Nintendo even eventually resorted to marketing third party games in an all-in-one commercial for Sonic Adventure 2, Agent Under Fire, and StarFox Adventures (which of course didn't come out for another half a year anyway).
2. Relatively big third-party support. I remember all the way through the first half of 2001, the Xbox shelves were always more packed than the GameCube shelves thanks to better third party support. Again, these weren't necessarily the best games, but all that mattered is that there WERE games. I think GameCube had maybe 3 or 4 games released in the first two months of 2002, most of which were unremarkable.
3. Halo...for whatever reason that game was huge, and it was able to keep generating interest well into the new year. While I think Halo was a little overrated, it goes to show the benefits of having a big flagship game. What was GameCube's flagship? SSB: Melee and Rogue Leader both seemed to be considered candidates, but neither game seemed to click that way at the time. I think Luigi's Mansion confused the issue, being the default substitute for Mario, but as much as I love it, there's no way it was a flagship title. I think things might have been really different if a game like Metroid Prime or even Super Mario Sunshine had been available at launch.
P.S. It's too late to change this now anyway, but a big part of Wii's success will depend on just how good the motion control is. If I feel like I'm really steering and controlling a car with my Wii, then why would I want to go back to ordinary video games?