No, the Wii unfortunately can't run games in resolutions higher than 480p, a sad fact for audio/visual connoisseurs. But in this fantastical age of technological wizardry and workarounds, that doesn't mean we can't have a glimpse of what could have been.
Using the Wii PC emulator Dolphin, YouTube user renebarahona (via Kotaku) has uploaded a few videos showing some Wii games running in 720p, including Super Smash Bros. Brawl (above) and Final Fantasy Crystal Chronicles: My Life as a King (below). Naturally, be sure to click the HD button to get the full effect. Not bad, huh?