Apple Inc. last year said it would invest $700 million to build the world’s biggest artificial-sapphire factory. On Monday, the company running that factory—GT Advanced Technologies Inc.—filed for bankruptcy protection.

The bankruptcy filing comes less than a month after Apple unveiled new iPhones with glass screens, rather than sapphire. Apple’s decision not to use sapphire followed tests in which the synthetic sapphire proved brittle, cracking when phones were dropped from various heights and angles, according to people...