After the games of September 5, 1964, the St. Louis Cardinals were tied with the San Francisco Giants for third place, 8.5 games behind the league leader, needing to pass the Giants, Cincinnati Reds and Philadelphia Phillies for the National League pennant and the only route to the postseason:
https://www.baseball-reference.com/boxes/?date=1964-09-05
However, the Cardinals, who went on to win the 1964 World Series, had 27 games remaining in the regular season.
After the games of September 12, 1995, the Seattle Mariners trailed the California Angels by 6 games in the AL West, and the New York Yankees by a half game in the Wild Card race, with only 13 regular-season games remaining:
https://www.baseball-reference.com/boxes/?date=1995-09-12
The Mariners finished the regular season tied with the Angels, beat the Halos in a one-game playoff to advance to the postseason, toppled the Yankees in a five-game AL Divisional Series before falling to the Cleveland Guardians in the AL Championship Series.
The 2019 Red Sox have remote chances of advancing to the postseason.