By the time Mark Bromley came to a stop, his strength had been almost entirely absorbed by the cold current pulling him downstream.

His waders were full of water. Fly fishing line was twisted and tangled around his body, and breathing was painful. All he could do was hold on to the root of an overhanging shrub while keeping his eyes, nose and mouth out of the water.

“I was just beat. Totally, totally beat,” Bromley recalled. “I couldn’t move.”

Such an outcome was not in the forecast for the day, which started out as an enjoyable fishing experience on the Madison River in July...