Nine days into the 2004 Tour de France, the U.S. Postal Service cycling team, led by Lance Armstrong, checked into a hotel near the village of Saint-Léonard-de-Noblat.

It was July 12, one of two rest days on the Tour—the rare breaks that give riders a chance to rest and gird themselves for the punishing climbs and sprints that make this the most depleting event in professional cycling.

According...