I have a custom object called Sports that ties to contact (student) information. The Sports object is used for coaches to review the qualifications of a prospective athlete (student). Since a student could be interested in more than one sport, they could have more than one Sports object. My question is - is it possible to have a related list on the Sports screen that would contain a related list to the additional sports objects?
What you are talking about is a "many-to-many" relationship. Each student can do multiple sports, and each sport can be done by multiple students.
The most natural way to implement this in Salesforce is through a junction object.
This will allow you to see the related lists and also supports reporting.
edit--- In the simplest example, your record would look something like this
The student has a list of as many sports as required, each student can re-use the same sports. The process of adding looks like this when you click on "new student sport"
and from the other side of the relationship, when you look at any given Sport you will see the students who are linked through the junction object, like this