It started with a sore throat. Brian German, 44, assumed he'd picked it up from one of his three children. But when it persisted, Brian examined the back of his throat and saw something that didn’t look right – a very large tonsil.

His other symptoms were mild – difficulty swallowing, a raspy voice. He attributed them to a job where he spends most of his time talking. But when the problem persisted, Brian saw a doctor. After...