Tara Flynn, a rising star at a TV production unit of 21st Century Fox, walked into her boss’s office last August and told him she was quitting and joining streaming-video giant Netflix Inc. The news was not well-received. “Netflix is public enemy No. 1,” said Bert Salke, the head of Fox 21 Television Studios, where Ms. Flynn was a vice president, according to a Netflix legal filing.

When Netflix finalized Ms. Flynn’s hire a few weeks later, Fox sued, accusing it of a “brazen campaign” to poach Fox executives. In response, Netflix argued ...