When Neo sees the Oracle, this is part of the conversation:
Oracle: But you already know what I'm going to tell you.
Neo: I'm not The One.....
Oracle: Sorry, kid. You got the gift, but it looks like you're waiting for something...
Oracle: Your next life maybe, who knows? That's the way these things go.
Neo: Morpheus...he, he almost had me convinced...
Oracle: Poor Morpheus...without him...we're lost.
Neo: What do you mean, without him?
Oracle: Are you sure you want to hear this?
Oracle: Morpheus believes in you, Neo. And no one, not even you, not even me, can convince him otherwise. He believes it so blindly, that he's going to sacrifice his life, to save yours.
Oracle: You're going to have to make a choice. In one hand, you'll have Morpheus' life..and in the other hand, you'll have your own. One of you is going to die..which one...will be up to you.
If he had believed he was the one, his attitude would have been different and he would have valued himself more than Morpheus, perhaps taking the attitude that they have to push forward, with or without Morpheus, as opposed to trying to save him - and it's a result of his trying to save Morpheus that he finally finds out he is The One.
Thinking he's not The One leads to a level of humility that we didn't see in Neo as "Mr. Anderson," before he knew about the Matrix, where he was cocky enough to give the agents, whom he thought was law enforcement, the "bird" by flipping them off with his middle finger.
If he did think he was The One, would that have changed his attitudes or behaviors enough that he could not do what he needed to do and would not have been The One?