Mike Lazaridis was home on his treadmill when he saw the televised report about Apple Inc.’s newest product. Research In Motion’s founder soon forgot about exercise that day in January 2007. There was Steve Jobs on a San Francisco stage waving a small glass object, downloading music, videos and maps from the Internet onto a device he called the iPhone.

“How did they do that?” Mr. Lazaridis wondered. His curiosity turned to disbelief when Stanley Sigman, the chief executive of Cingular Wireless joined Mr. Jobs to announce...