Most of what we know of this period in Japan comes to us from 2 works, the Nihon Shoki which serves as the official state history of the Yamato state up until that point, and the Kojiki which is largely a collection of poetry of historical value due to the fact that it was mostly the aristocrats and soldiers who were composing them. What you need to understand first about the Nihon Shoki is that it's essentially propaganda. It's main purpose is to lend authority to the imperial line of the Yamato state by associating their personages with the gods of the people. It isn't necessarily reliable, much like numerous aspects of the various chinese state histories. And as a matter of fact, there are reasons to doubt aspects of some of the older sets of poems in the Kojiki. In the Nihon Shoki it says that Jimmu is a descendant of the sun goddess Amatarasu and the storm god Susa-no-O. Now even though both the Kojiki and Nihon Shoki corroborate his reign dates, they were both commissioned works from the aristocracy leading the Yamato state in (probably) Asuka and they were commissioned at about the same time anyway, and after several hundred years after Jimmu lived. If you take it for granted that he existed in the first place, you may accept the secular information about him while taking a grain of salt over his deeds as depicted in the Nihon Shoki. You certainly can't have any faith in the mythical information about him or the absurd, such as him living for 126 years. If he existed at all, that tells you that the following reigning emperors do not have the correct reign dates, as well for Jimmu since he could not have lived that long. However, the point of the Nihon Shoki is to present him and the imperial line as divine. As a result of this, we really don't know how far the imperial line goes back, because what we do know isn't trustworthy here. Jimmu not existing in the first place is certainly possible because the writers of the state history had a political need of such figures anyway. Alternatively the legend of Jimmu may have predated the state history and just been very convenient for the fledgling Yamato state. The only way to know for sure if he really existed would be for the discovery of a kofun burial mound that clearly identifies him by name and can be carbon dated to the time period, but so far that hasn't been discovered (to my knowledge). I would say that given the current knowledge we have, it's a safer bet that he never existed in the first place, but we can't know for sure. Large sections of history come to us from a few biased sources and this is just one of those cases.