First of all, thanks to Jess1 for the reference to our website!
IsItJustaGame?, to follow up on what Jess1 stated, the best chance for a waiver request to be approved is when the circumstances are "beyond the control of the student-athlete or coach." So, as pointed out, a season-ending injury or serious illness incurred by the athlete and supported by required medical documentation might be grounds for the NCAA to approve a waiver request.
If a counter quits the team, becomes ineligible, or is cut from the team, it would be highly unlikely for a waiver request of that type to be approved. In addition, the waiver request must be submitted by the school. An athlete can't submit a waiver request on their own.