When she was in her early 30s, Katy Mathes decided to check her cancer risk. A genetic test showed a mutation on a BRCA gene, which significantly raises a person’s lifetime risk of developing hereditary breast or ovarian cancer.

Thirteen people in the family got tested—her mother, her sister, cousins and aunts. Eleven had the mutation. Almost all did their testing with Myriad Genetics Inc., which introduced the first BRCA tests in 1996.

“I...