The intern took it upon herself to place an inscription above the entrance to the Geek Holocron archives: “Whosoever holds this holocron, if she be worthy, shall possess all geeky knowledge.” Thankfully, she ‘inscribed' it in burnt sienna Crayola; she said it looked older and more rustic that way. I said it was a good thing crayon washes off stone so easily.
I have been rewatching the Marvel movies in anticipation of Captain America: Civil War when I got to [Avengers:] Age of Ultron and I wondered if we ever found out why Captain America couldn't lift Thor's hammer. Was that ever made known? You'd think if anyone could it would be him.
For those who don't remember, this is the scene after the big party at Stark Tower. All of the Avengers are there, with the addition of Maria Hill, and Tony Stark mocks the inscription on Mjolnir that claims whoever is worthy may wield the hammer and claim the power of Thor. Being men with egos, they attempt to lift it all in turn; even Stark and Rhodey in their Iron Man and War Machine gauntlets can't budge it. Yet when it comes to Steve Rogers, he makes it shift just the smallest bit. He still can't lift it, but he gets further than anyone other than Thor, and later Vision. (You can watch the scene here.)
Fans of Marvel comics will know that Captain America has wielded Thor's hammer a few different times, so at least on those occasions he has been deemed worthy. There are likely a few story reasons why the Marvel Cinematic Universe version of Captain America can't quite lift Mjolnir, but let's take a look at some of the theories that have floated around in the year since Age of Ultron came out.
One of the main theories is that Cap could have picked it up, but he decided not to when he noticed it budge and saw Thor's face drop. As the theory goes, Steve realized that Thor put a lot of weight behind himself being the only one worthy of wielding such a magnificent hammer, which is fair considering all he went through in Thor to be able to lift it in the first place. Because so much of Thor's self-esteem and self-worth came from being the only one worthy enough, Steven didn't want to take that away from him and only pretended to keep trying once the hammer started to give way.
Just recently, YouTuber Desmond Drake posted a video where he claims Captain America's biggest flaw, and the one thing preventing him from picking up the hammer, is that he is a soldier above all else. As he says in the video, “He only finds worth and purpose through fighting.” He also points out that Thor had to learn how to avoid conflict when necessary in order to wield Mjolnir again after it's tossed into the New Mexico desert. However, Cap has also said that he doesn't enjoy fighting; he just doesn't like bullies.
Another big theory is that Steve Rogers can't lift Thor's hammer not because he isn't worthy but because he doesn't think of himself as worthy. It's easy to forget that for the first twenty-one years of his life he was a warrior in the body of a weakling, and he's not yet even thirty, so he likely thinks of himself in his deepest parts not as the perfect human specimen but as a small man whose body consistently fails him. This is something Steve has expressed in the comics, so it's not outside the realm of possibility that he has placed a mental block on himself that simply prevents him from truly picking up Thor's hammer. Feeding into this theory is the storyline of Captain America: Winter Soldier. Cap likely feels that he's failed Bucky, failed to fully eliminate Hydra, and put the lives of countless Americans at risk. With all of that in his recent past, it would be surprising if he didn't doubt himself.
Probably the most likely answer is that Cap simply had no need to wield the hammer at that point in time, so though he's worthy, there was simply no need for him to lift it. Of the handful of legitimate times Mjolnir has been picked up by someone else, they have always been a time of need. Even the current incarnation of Thor, Jane Foster, was only able to pick up the hammer and become Thor because Nick Fury whispered something secret in Thor Odinson's ear that made him unworthy. It's not outside the realm of possibility that if the time comes in the heat of battle, like during either the first or second part of Avengers: Infinity War, Cap will be able to use Mjolnir while Thor is incapacitated. Alas, we'll need to wait until 2018 and 2019 to find out for sure.
One of the fun things about this question is that we still don't know the answer. Until someone involved in the film—a screenwriter, Chris Evans, Kevin Feige—gives us a definitive reason, we can continue to speculate.
Bonus Trivia: There have been a few times when someone unworthy has managed to lift Mjolnir, but each time has been a bit of a cheat. Magneto's powers have sometimes worked on the Asgardian metal that forms it, uru, but that's depends a lot upon the writer. Another notable cheat was when Red Hulk, back when he was still a bad guy, tossed Mjolnir and Thor back to Earth from the weightlessness of space. It wasn't so much that he picked it up as there was no perceptible gravity to affect it.