Are you allowed to give your students assignments that involve redesigning existing brands?

I did not go to school for design, so I really have no idea if this is allowed, but what would happen if you told the students that their assignment was to create an ad for Nike?

I'd jump at something like that.

It's, in fact, fairly common. Some companies even go so far as to ally with the AAF and other organizations for national competitions.

Here's some example student projects from a leading design school:

http://www.artcenter.edu/accd/gallery.jsp

Refresh a couple times to see more. Many are commercial, and some of those are for established brands.

It's not the assignment that matters, it's the initiative that matters. Giving students the assignment to do something awesome isn't any different from giving them an assignment to do something ordinary.

Or if you can't do that, tell them to invent a competitor and brand that?

I don't think that works as well. Doing an ad for nike feels cool, and you get to brag about it on your facebook page.