Why can't Morpheus, Trinity, etc. behave like the agents as Neo does?

Neo has within him a portion of the Source Code of the Matrix. This grants him all his abilities. That is why he is referred to as the One. Only the individual with the Source Code can match, or beat the Agents, fly, stop bullets, and perceive the Matrix as code. In the Matrix Reloaded the Architect says "The function of the One is now to return to the source, allowing a temporary dissemination of the code you carry, reinserting the Prime Program."

Because Neo is The One.

The One, which is also known as the Prime Program, is a systemic feature of the Matrix, the code for which is carried by a human being who possesses vast superhuman abilities as an avatar within the Matrix in addition to some extraordinary powers in the real world.

Why the Matrix bothers people inside it? Like a splinter in your mind - why is it feels like this?

The Matrix is shown as a world full of limitations and controls, most of which feels artificial to ones similar to Neo. They are without explanation. Neo doesn't like this.

Does Neo, Morpheus and the others reject the Matrix because the rules are without explanation? I think not, they reject it because it bothers them that they are without explanations or that they exist at all.

They all have something in common - they are uncomfortable with the Matrix because it limits them, something inside them.

Neo: I have these memories from my life.... None of them happened.
... What does that mean?
Trinity: That the Matrix cannot tell you who you are.
Neo: But an oracle can?
Trinity: ... that's different.

It's latin. It says temet nosce.

What the Oracle does to Neo is called different from what the Matrix does to him, because the Oracle encourages and helps him to "know thyself", to explore what is already inside him, the same thing that was limited by the Matrix.

Morpheus: Do you believe in Fate, Neo?
Neo: No.
Morpheus: Why not?
Neo: Because I don't like the idea that I'm not in control of my life.

Morpheus knows that there is something inside Neo trying to be awaken, but it's up to Neo whether he chooses to awaken it or not.

And this something is unique, and it's directly tied with Neo. He is beginning to unroll it when he is beginning to believe in it. I think they choose their names after it.

Neo is the only one who we follow from being an slave of the Matrix ("bluepill") to being free ("redpill") to undergo a combat training to become sort of a warrior inside the Matrix. The process he goes through seems to be a sort-of a routine one ("You did all this?", "We're supposed to start with these programs first.", "No one's ever made their first jump.") and Neo only gets to his (expected) full potential when he is beginning to believe.

Believe in what? I think he begins to believe that he is the One. In other words, and from an other perspective, he begins to believe in himself on a greater level (which he by then perhaps perceives as the real one, in contrast with some of his previous beliefs).

So if you think about it, the power of Neo comes from his ability or harmony to believe in himself. But this follows that this belief is tied to what you perceive as your "real" self - and I believe Morpheus and Trinity has different views about what they saw as a role for themselves in life.

This is reflected with what the Oracle tells each of them:

Neo: What did she tell you?
Morpheus: That I would find the One.
...
Oracle (to Neo): But you already know what I'm going to tell you. [... ...] You got the gift... ...but it looks like you're waiting for something.
...
Trinity: The Oracle told me... I'd fall in love, and that the man I loved would be the One.

Neo achieves advanced levels in combat training and have the ability to use the Focus (as their concentration skill is called in the game Enter the Matrix). But he can only unroll his exceptional abilities when he is "beginning to believe" in himself.

So I think that Morpheus and Trinity doesn't free their mind to the "extent" -or rather, the way- Neo does, because they can't imagine themselves in a position similar to Neo's.

(Note that, while it's out of the scope of the movie The Matrix, Neo is represented in the follow-up movie Reloaded like his extreme capabilities actually give him great challenges on what to use them for and why, as he expresses frustration at the beginning of the movie.)

The other ones cannot tie their beliefs with themselves like Neo does, because they are not "the One" - maybe because they do not wish to be the One (for whatever reason).

Now why don't they simply learn this ability from Neo, once he found out the way?

Neo may have different limits than the others, based on his real self - "10 hours straight"! Tank mentions it that Neo is "like a machine". It is possible that the others cannot keep up with his pace. He advances too fast - the others won't get to his level any time soon.

Also I think this is not something you achieve by training. The other people have incompatible souls. Neo uses, utilizes his personality to the extreme. You could almost say he "exploits" his personality in order to achieve the Matrix-altering effects. Since all of the team rejected the Matrix because they didn't want external power tell them who they are, they are not going to change themselves. They cannot believe in themselves as someone they are not.

However, they can believe in each other. Reloaded further explores this. But already in The Matrix, becoming the One was a "group effort", as Carrie-Ann had put it.

If you view it from this perspective, you can sort-of think about it like "it requires multiple person's work to put together the skill-set of a single One". I think this is not only technical, as Neo really have to believe that he is the One in order to be "in the right mindset". Notice that this is not just technique. (And Morpheus already told him earlier that his weakness is not his technique.) To be the One requires events that are like stepping stones for your mind.

The humans cannot and wouldn't want to simulate the events needed by Neo to become the One. For example, they wouldn't want to find a traitor who kills most of their teammates for every one of them. So it is not possible to simply train all of the tea to become another One.

On the other hand, in Reloaded, both Morpheus and Trinity, those closest to Neo, do fight agents. Only Morpheus attempted that with old agents and only to win time. Now they do both do it with greater success. Even if those are updated agents. It must be that Neo taught and trained them as much as humanly possible. Both their minds and technique.

I'm surprised that no-one has brought up the most relevant quote in my opinion to this discussion, which is the first thing that came to my mind:

Trinity: How did you do that? Neo: Do what? Trinity: You moved like they do. I've never seen anyone move that fast. (M1)

This quote takes place in the first of the matrix movies. As a viewer we have a basic introduction to Neo possibly being the "One" but this is in no way confirmed. We haven't learned by the

Maybe I misinterpreted what the architect said, but I believed (I will have to check on this) that the appearance of the One is something that happens not because it is coded into the matrix, but that it because of human being's inability to accept an imperfect matrix, i.e. the robots do not know how to make a matrix that humans will accept because they do not understand humans that one of the human being's avatars in the matrix will gain super-ordinary powers and will keep growing in them until a final matrix destroying battle takes place.

It's like a physics model of equilibrium where if there is even a tiny chance that it will lose its state of equilibrium given enough time it will have to give in to this error, even if it is only .0001 % or something.

I think the robots, and the programs, make it pretty clear that they want the matrix to continue on indefinitely without the need for this restarting, and that each time they restart they use the information that caused the previous failure and try to make a better matrix. Basically it just pisses the robots off that humans will not be happy, a sort of ironic philosophy coming from a computer program. This is also what Smith tells Morpheus in the first movie:

"I hate this place. This zoo. This prison. This reality, whatever you want to call it, I can't stand it any longer. It's the smell, if there is such a thing. I feel saturated by it. I can taste your stink and every time I do, I fear that I've somehow been infected by it."

Basically he wants out, or he wants to know more because of his self-awareness, the other Agents think he is a pretty odd program. As a program, or ghost in a machine, he feels desire when he takes his earpiece out and talks to Morpheus privately. Smith doesn't feel it's weird to act like a human, Neo doesn't think it's weird to move like a machine, everybody else thinks they are crazy. Problem solved.

The people in the Matrix live in a virtual world. They are not real people, they are only simulations. To function in the Matrix, every avatar has a set of data and instructions that defines how it works. In object-oriented programming, that set of data and instructions is called a class.

There is a basic class that defines human avatars. To preserve the illusion of their virtual reality, humans have limits. Those limits may vary from person to person, but some are absolute. For instance, just as in the real world, a human can't fly without some kind of assistance.

Agents aren't human, and they aren't avatars. They have a different set of limits. They are faster and stronger than humans because that's the way their class is defined. Theoretically, an avatar could move at the speed of light, even move from one place to another instantly, if it had no limits.

Neo is different. His class is different. His avatar has different limits. He can fly, and he can hold his own against an agent. The other characters are still human.