The Nationals have only a one percent chance of making the playoffs. Has any other team ever conquered such long September odds and made the playoffs?
Yes. On September 14, 1964, the St. Louis Cardinals were six games behind the Philadelphia Phillies and had only a 2-3% chance of making the playoffs. (In 1964, there were no divisions and no wild card playoff spots.) In other words, the Cards then were in a position much like the Nats are today. In one of the greatest collapses in baseball history, the Phillies went 6-13 for the rest of the season and between September 21 and 30 lost ten straight games. Needing a win and a Cardinals loss on the final day of the season to make the playoffs, the Phillies won but missed the playoffs because the Cardinals won also.
Can it happen again? Yes, but it's not bloody likely!