Something that's intercollegiate occurs between different colleges. At an intercollegiate debate competition, there are teams from many schools competing against each other.
You'll almost always find the adjective intercollegiate describing sports, or occasionally scholarly rivalries, between colleges or universities. An intercollegiate baseball association is made up of teams from many different colleges, and an intercollegiate lacrosse tournament also includes a variety of schools. Intercollegiate is made up of inter-, "among or between," and collegiate, from Medieval Latin collegiatus, "pertaining to a college."