"There are so many moments in life when people don't say what they mean, when they are missing each other, waiting to run into each other in a hallway." - Sofia Coppola.
As much as we want our cinematic heroes to go above and beyond, we also empathize with them when they do not -- when they do not take the risk, speak the words their heart desires for them to say, admit the one thing that would save a dying relationship, do the one thing that could save their life. Within this basic concept, we can understand the makings of an Anti-Climax.
In Constructive terms, we define an Anti-Climax as a disappointing end to an exciting or impressive series of events. One famous example is the sudden death of Josh Brolin's character in No Country For Old Men -- a character we come to believe, after going to such great lengths, will defeat his antagonist, however impossible an endeavor this may seem.
Notice how, while Brolin's death does fit into the definition above, that definition does not fully encompass this moment on a level of meaning. In other words, one might ask themself, "Why in the world would the Cohen Brothers want to do something so disappointing in their screenplay after a series of such impressive events?"
On a level of meaning, though we would have all loved to see Brolin defeat Anton Chigurh, for him to do so would have taken away from the film's ultimate climax born of the questions of fate -- when Chigurh, a man who's determined the life and death of other characters throughout the film, is in a violent and unexpected car accident.
You see, the thrill, the exhilaration, the climactic nature of No Country For Old Men exists within the higher powers of chance that can take our lives at any minute, change the course of history, or the direction of society. As the famous interaction goes in that family-owned gas station scene midway through the film, Chigurh points to and foreshadows this concept beautifully when he says, "You've been putting it up your whole life. You just didn't know it." In other words, we are always risking our lives with every decision we make.
It is this that makes an Anti-Climax so powerful and so resonant. It proves a climax. It is a tool for a climax's solidification in the reader or the audience.