After working at Theranos Inc. for eight months, Tyler Shultz decided he had seen enough. On April 11, 2014, he emailed company founder Elizabeth Holmes to complain that Theranos had doctored research and ignored failed quality-control checks.

The reply was withering. Ms. Holmes forwarded the email to Theranos President Sunny Balwani, who belittled Mr. Shultz’s grasp of basic mathematics and his knowledge of laboratory science, and then took a swipe at his relationship with George Shultz, the former secretary of state and...