Two former Theranos Inc. directors said they didn’t follow up on public allegations that the Silicon Valley blood-testing firm was relying on standard technology rather than its much-hyped proprietary device for most tests, according to newly released court documents.

In depositions, the highly decorated former directors—former U.S. Navy Adm. Gary Roughead and former U.S. Secretary of State George Shultz—who were board members when concerns of employees and regulators became public—said they didn’t question Theranos founder...