Yes, the Cardinals have won a pennant before when they’ve been almost this far behind the leaders.
In 1964, they trailed the Philadelphia Phillies by 11 games on July 8 and roared back to win the National League title on the last day of the season.
In 2011, they trailed the Milwaukee Brewers by 10½ games on Aug. 27 and were 10 games out of the wild-card race on that day. They didn’t win the division title but they did run down the Atlanta Braves for the wild card and proceeded to claim their most recent World Series title.