On Theranos Inc.’s website, company founder Elizabeth Holmes holds up a tiny vial to show how the startup’s “breakthrough advancements have made it possible to quickly process the full range of laboratory tests from a few drops of blood.”

The company offers more than 240 tests, ranging from cholesterol to cancer. It claims its technology can work with just a finger prick. Investors have poured more than $400 million into Theranos, valuing it at $9 billion and her majority stake at more than half that. The 31-year-old Ms. Holmes’s...