Ari Bernstein was 22 years old and five months into his first semester at Albert Einstein College of Medicine when he learned he was sick. Sometime last November, he began noticing his neck was sore, and felt a lump at the base of his neck. When he had it checked in December, it turned out to be the tip of a roughly seven centimeter mass wrapped around his heart and trachea.

Doctors wrestled with the diagnosis: Was it a rare cancer that would require major surgery or was it lymphoma, a cancer that was serious but generally...