That's right. I suppose if it's just part of a university course, copyright wouldn't be a big concern, but it would make any public exhibition virtually impossible to do legally. That could be a serious disappointment if it turns out well. Depending on how many different movies are used, acquiring the necessary licensing to do it legally would be a huge job, and potentially expensive. Just getting the rights to show a movie can be a pain, but that would be nothing compared to getting permission to cut together bits of different movies from different owners to make a new movie.
Again, if there's no chance of a public showing, I wouldn't worry about it. But I think it would be sensible to do as Flick James says, and use public-domain movies. In addition to features, there are loads of short educational films in the public domain. I think staying in the public domain would also give the finished movie a fun kitschiness, due to the age and nature of many of the films.
You can find loads of stuff here:
http://www.archive.org/details/movies