Students cannot remove or hide a course from their course list. However, if they officially drop a course, they will be disabled in the enrollments and will no longer see the course listed.
Course completion does NOT impact course visibility.
Once Blackboard term expires, about 3-4 weeks after exams are over, students should see the course listed, but may no longer have access unless 1) the instructor has closed the course (e.g., made it unavailable) or marked it complete. However, if an instructor modifies the course properties to change the course end date or to remove it altogether, the course may still appear as current.