The Architect says something about the Neo's predecessors but gives no explain about them. Was Dan Davis one of the first Ones? He also learned to see the code, woke himself up, and bent the space of the Matrix (for example, he seems to fly a bit). But was he really the One or simply, as the movie states, someone with exceptional perception?
No. The function of the One requires that the One meet with the Architect and make the choice between rebooting the Matrix or causing the extinction of the human race (by allowing both the Matrix and Zion to be destroyed). All of Neo's predecessors met with the Architect and chose to reboot the Matrix. Dan Davis did not escape from the Matrix much less meet with the Architect to make the choice of the One, so he could not have been one of Neo's predecessors.
Although Dan Davis can bend the rules of the Matrix, so can any red pill (Morpheus, Trinity, Cypher, etc.). The One is able to bend and break the rules of the Matrix more than other red pills, but this is only a difference in degree. The most unusual behavior exhibited by Dan Davis was a short self-substantiation (when he saw his pod in the real world without the use of a red pill), but he's not the only one to have done it -- the Kid also self-substantiated but the Kid was not the One because Neo was the One at that time.
Dan isn't the One. He's simply someone who gains exceptional perception, resulting from his enormous mental exertion. As explained at the start of the short, he's not one of those who reject the Matrix (like Morpheus and Trinity) and he's certainly not one of the Oracle's 'potentials'. He's something else.
VO: Only the most exceptional people become aware of the Matrix. Those that learn it exists must possess a rare degree of intuition, sensitivity and a questioning nature.
However, very rarely, some gain this wisdom through wholly different means.
This man is one of those few.
We see a very similar set of circumstances in the Matrix Webcomic "Sweating the Small Stuff". A small time hoodlum named Dez, fuelled by a combination of good drugs and bad stress begins to sense that something is amiss.