The night before a big meeting with a Swiss drug company in 2008, Theranos Inc. founder Elizabeth Holmes and a colleague sat in a Zurich hotel, sticking their fingers with a lancet.

They drew drops of their own blood to try the company’s testing machine, but the devices wouldn’t work, says someone familiar with the incident. Sometimes the results were obviously too high. Sometimes they were too low. Sometimes the machines spit out only an error message.

After...