Apple Inc. executives had every reason for optimism when they approached Walt Disney Co. in early 2015 to join the streaming television service Apple planned to launch. Disney Chief Executive Robert Iger is an Apple director and had said he was keen to strike a deal.

Disney, which owns channels such as ESPN and ABC, was stunned, though, when Apple executive Eddy Cue made demands that would have upended decades of cable-industry and Hollywood practices, people familiar with the discussions say.

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