On July 5, 2005, the Nationals beat the Mets 3-2 to bring their record in one-run games that year to an incredible 23-7. At the time, there was a lot of praise heaped on Frank Robinson's late-game strategy, the Nationals' bullpen, and the team's overall ability to "find ways to win"--a lot of the same points you're hearing now.
By the end of the season, the team's record in one-run games was 30-31.
What happened? Did the Nationals' bullpen fall apart? Not really. Chad Cordero's ERA rose from 1.13 in the first half to 3.04 in the second, but that's hardly cratering. Gary Majewski's got better, falling from 3.21 to 2.66. Luis Ayala: 3.06 to 1.15. Hector Carrasco: 2.43 to 1.80.
Presumably the Nationals didn't suddenly lose the intangible "scrappy winner" quality we heard so much about. Frank Robinson was always a terrible in-game strategist, so there's not much point in trying to break down that particular constant.
Is winning close games a repeatable skill at all? If it is, then you'd expect the best teams to have it. Last year, the eight playoff teams were the Yankees, Red Sox, Dodgers, Rockies, Phillies and Cardinals were worse in one-run games than they were overall. The Angels had an almost identical winning percentage. Only the Twins were better. Those were all teams that "knew how to win." Most of them had pretty good bullpens and managers.
Of course, we'd need a much bigger sample size to conclude that winning close games is definitely not a repeatable skill, but someone else will have to do it. But it's certainly a questionable proposition. Nationals fans should be highly skeptical that the Nationals' good fortune in nail-biters will continue.