There's a really good series on Youtube with a similar premise, Untitled. It focuses on individuals but in the process, discusses certain dynasties that never quite reached the peak of their sport.
For me, there are two particular runs by the Vancouver Canucks that I remember being achingly close to titles: two years during the West Coast Express era, and then the actual trip to the finals in 2011.
2002-2003, Vancouver rolls into the playoffs with swagger, led by 104 points from Markus Naslund, 144 PIMs from Todd Bertuzzi, and Brendan Morrison was on their line too. In the first round they face some adversity, going down 1-3 to St. Louis, but then they rally back and win three straight to take the first round series in 7. With momentum on their side, the Canucks go up 3-1 to Minnesota, and then collapse inwardly, dropping that series by losing three straight after winning three straight the previous round.
But the core of the team is still going strong, although Naslund plays hurts and only gets 84 points, BMo and Toddbert The Uzi both add 60, and some kids called the Sedins (obscure Swedish dudes) have a flash in the pan season and each add 50 points from the second line. As the team looks forward to the playoffs, disaster strikes as Todd Bertuzzi loses his fucking mind and tries to decapitate Steve Moore. So Bertuzzi gets a years suspension, and in the pivotal game seven, late in the game, Ed Jovanovski is in front of the net, trying to be Todd Bertuzzi.
So Jovanovski gets called for the penalty, but four on four, the Canucks score, to tie the game and send it into overtime. The problem being, Jovo is still in the box, and in the overtime frame, Calgary scores on the powerplay, and advance to the next round. The West Coast Express slowed down hard after that, and was never the same.
As for 2011, that's a more familiar tale, so I'll not rehash it.