That's a tough one, Jennifer. The University of Wisconsin was established in 1849, and the badger has been the crest of Wisconsin's coat of arms since 1851. Football became an official sport at the UW in 1889, and shortly thereafter adopted the badger as its mascot.
But it wasn't until 1940 that Bucky Badger as we know him today, sporting a red and white letter sweater, was brought to life by artist Art Evans. At the time, the badger went by several names including Buddy, Bernie and Bouncy. But in 1949, a contest was held on campus to officially name the badger mascot, and the winner was “Buckingham U. Badger.”
While the University of Michigan-Ann Arbor does not have a wolverine mascot on the football field, students have been calling themselves “wolverines” since 1861 — nearly 30 years before Wisconsin founded its football team.
According to the University of Michigan-Ann Arbor's “History of the Wolverine Mascot at UM” Web site, after seeing Wisconsin bring live badgers onto the football field in 1923, Wolverine football coach Fielding Yost decided that his football team needed a mascot and set out to find one. But while real-live badgers abound in Wisconsin, no wild wolverines exist in Michigan, so Yost couldn't find a live mascot. He then decided that any wolverine, dead or alive, would do, and eventually got word that Michigan Senator William Alden Smith owned a mounted wolverine. A visit to the senator's home unveiled that the wolverine was actually a coyote.
During the 1990s, the University of Michigan had a mascot named “Willie the Wolverine,” a modern mascot in a suit who cheered for the football team. But that attempt was short lived, and there is no longer an athletic wolverine mascot at UM. So the answer to this question is all in how you look at it.