When the Oracle talked with Neo, she said he wasn't the one YET, that he was waiting for something; "Maybe your next life, who knows". And Neo doesn't really become the one until his next life (after he dies).

Also, Neo isn't the first one. It seems to me that there are links between being the One, and being the Buddha (note that the kid who tells him "there is no spoon" is often call the Buddha Child). Both the One and the Buddha are defined by a state of mind, realizing the truth about the world.