Why didn't Harry ever learn to cast spells without vocalizing? Hermione learned how to do it, and Snape warned Harry about learning how to do it. It would have helped in many places.
In Half-Blood Prince Harry effectively incants Levicorpus non-verbally on Ron; Ron ends up hoisted into the air, as if held up by the ankle. Harry then does a non-verbal incantation of Liberacorpus, which releases Ron. (Half-Blood Prince - Chapter 12)
Following Dumbledore's death, Harry chases Snape and the Death Eaters across Hogwarts' grounds to the area around Hagrid's hut, where Harry attempts to duel Snape. He casts his spells verbally, except for once, where he "mustered all his powers of concentration" and tried to again cast Levicorpus on Snape non-verbally. However, Snape easily blocks Harry's attempt.
Harry attempts to curse Snape multiple times, but Snape deflects all of Harry's spells. Snape says, "Blocked again and again and again until you learn to keep your mouth shut and your mind closed, Potter! (Half-Blood Prince -- Chapter 28)
In Order of the Phoenix, Harry is shown to be a very poor Occlumency student (i.e. he cannot block Snape from reading his mind using Legilimency) and Snape continuously degrades Harry for being too emotional, too weak, and too unskilled to perform Occlumency. Occlumency is also a non-verbal skill.
Harry is generally characterized as sometimes not having control over his mind and/or emotions; self-control over the mind, concentration, and emotions is necessary to perform non-verbal magic. It would seem that Harry just needed longer to progress to the point of mastering non-verbal spells; as he matured, he likely gained better control over his emotions and mental strength. After all, he did go on to head the Auror Office at the Department of Magical Law Enforcement.
"Yes, those who progress to using magic without shouting incantations gain an element of surprise in their spell-casting. Not all wizards can do this, of course; it is a question of concentration and mind power which some" -- [Snape's] gaze lingered maliciously on Harry once more -- "lack." (Half-Blood Prince -- Chapter 9)
Some might argue that Harry successfully gaining access to the Room of Requirement would demonstrate some ability of non-verbal magic. In order to get into the Room of Requirement, one must stand in front of the wall where the entrance is located and think of exactly how they need the room to function, and that they have genuine need to use it to begin with.
He did; it was required when Snape taught DADA; it simply wasn't one of Harry's better skills. I don't have a copy nearby, but I believe Harry even used some on Snape in The Half Blood Prince, but was foiled by Snape using Occlumency... He countered him again and again and would do so until he 'learned to close his mind' as I recall.
I believe he also used Levicorpus non-verbally. See: the Nonverbal Spell entry on the Harry Potter Wiki for a list of known uses.
In the movies he fails simply so that people can hear what is being done rather than just have sparks fly everywhere and in the books it is used to make him vulnerable, like kryptonite to superman. A Harry Potter able to do nonverbal spells would have made him one overpowered mofo and could have just won every duel and battle until the end of the 7th book. Not being able to makes him vulnerable and "human" as well allowing for dramatic affect and in the final battle with Voldemort it allows for Harry to say Expelliarmus and Voldemort to say Avada Kedavra, and makes it epic, connects their wands and so on. Nonverbal Expelliarmus would not have had the same effect, especially in the movies.