I guess it's possible, but that sounds fishy. Why would they delay the plane for a passenger who feels sick? Either you feel fine and you fly, or you don't and you get off the plane. You would never delay a train or a bus for that reason.
If the plane is still on the ground and a passenger says they are too sick to fly, the plane goes back to the gate to let them get off the plane. That's normal.
A passenger said they were too sick to fly, so the plane went back to the gate. When it reached the gate, the passenger changed their story and things got more complicated.
Also, imagine if you complained that your stomach was hurting and you had to get off the plane. If they didn't turn around on the tarmac and then your appendix bursts or something then you or your next of kin owns an airline when the lawsuit is over.
