We know the names of Olympic victors as far back as the first (recorded, the tradition according to Eusebius is that there were 28 Olympiads before 776) Olympiad, in which Eusebius tells us an Elean named Coroebus won the στάδιον footrace, then the only event included in the games. The Olympic victors were a big fucking deal, and their names were memorialized, as often were their images. Olympic victors had victory odes written about them, most famously Pindar's, and they often became by virtue of their glory and influence pre-eminent statesmen or envoys--at Sparta Olympic victors marched beside the kings in battle. Their names were committed to poems, inscriptions, the bases of their statues, and coins, and great care was taken by the city of a victorious athlete to bring proper attention to his (and his city's) success. We know of around a quarter of the victors at the Olympic festival, far more than we know about the victors of the lesser festivals of the athletic circuit. Olympic victors were maintained on official inscriptions, including one that we have that lists the names of victors for 400 years (if I remember right it's Hellenistic). Besides that and other inscriptions, such as those set up by cities or on monuments and tombs, we have hundreds of textual references to Olympic victors. Pindar is basically all athletic victors, predominantly the Olympic victors, and there's no shortage of authors who give us the names of Olympic victors. There are also papyri, including the famous Oxyrhynchus Papyrus 222, which allowed us to date several victory odes by using the list contained in the papyrus, but like all papyri good-sized scraps are hard to come by. Where our lists overlap there have never been any problems (importantly Papyrus 222 tells us that even in 3rd Century Egypt the names of early Classical victors were being preserved and were probably pretty easy to check), and given the literal sanctity of the Olympic victors list it seems hard to believe that the official lists were easily or often falsified--in any case, had one city or individual tried to falsify their copy there were plenty of other lists, and the official inscribed lists were always available at Olympia