Many Christians in the 2nd century were arguing for what’s called “Apostolic Succession” – that they were in a direct line of teachers who were taught by Jesus. Valentinus claimed that he was a student of a “Theudas” who in turn was a student of Paul. Irenaeus (who names the gospels, and thus, is where current Christianty is descended from) claimed to be a student of Polycarp who in turn was a student of John. Eusebius (writing his “Church History” in the 330s CE) claims that Papias, who was a contemporary of Polycarp, knew some people who knew the apostles themselves; and that a “Mark” was a disciple and interpreter of Peter.
Valentinius lived around the middle of the 2nd century. Irenaeus also lived around the middle of the 2nd century.
In the “Martyrdom of Polycarp”, Polycarp claims himself right before he dies that he served Jesus 86 years (if he was martyred, then who wrote this story?!), which some people date to being around the year 160 CE. So Polycarp was such a menace as an 86 year old that he had to be stabbed to death (after people tried to burn him but miraculously didn’t get harmed by the flames). So this means that Polycarp was born sometime around 75 – 80 CE.
In 70 CE millions of Jews are killed or starved to death because of the first Jewish-Roman war (millions more died in the Bar Kochba Revolt).
So if Jesus had disciples, and we don’t really know if any of them were older than him, we’ll assume that they were around the same age as Jesus. So that makes a whole bunch of 70 year olds during the first Jewish-Roman war (if they weren’t Jews then I guess the War doesn’t really apply – but then again, there isn’t any “tradition” of Jesus having Gentile disciples, and in Matthew he explicitly tells his disciples not to go into any Gentile or Samaritan towns, so…). Polycarp wouldn’t have gotten “lessons” from any disciples until maybe 10 or 15, so that adds an extra 15 years to Polycarps’ birth – 95 CE. So in order for any Apostolic Succession to work, we have to assume that a 90 year old taught a 15 year old and then this 15 year old also lived to be a 90 year old and taught Irenaeus or Valentinus. Both of those successive 90 year olds were poor itinerant preachers travelling all around the Roman Empire evangelizing. Without money or access to the first rate medical care that the upper class Romans had (who they never evangelized to, since Christian targets of evangelism were the uneducated lower classes).
But Polycarp also was not only instructed by apostles, and conversed with many who had seen Christ, but was also, by apostles in Asia, appointed bishop of the Church in Smyrna, whom I also saw in my early youth, for he tarried [on earth] a very long time, and, when a very old man, gloriously and most nobly suffering martyrdom, departed this life, having always taught the things which he had learned from the apostles, and which the Church has handed down, and which alone are true.
To these things all the Asiatic Churches testify, as do also those men who have succeeded Polycarp down to the present time,— a man who was of much greater weight, and a more steadfast witness of truth, than Valentinus, and Marcion, and the rest of the heretics. He it was who, coming to Rome in the time of Anicetus caused many to turn away from the aforesaid heretics to the Church of God, proclaiming that he had received this one and sole truth from the apostles—that, namely, which is handed down by the Church.
– Irenaeus, Against Heresies 3.3.4
All Christians in the 2nd century were telling these sorts of “tales” to authenticate their churches. We just are descendants of the ones that won.