I doubt any Second Century Christians ever knew an Apostle. Its the time thing.
Ignatius of Antioch was an elderly man when he was carried from Antioch to Rome. There is a pretty solid consensus that Ignatius wrote his letters (which were written on the way from Antioch to Rome) sometime between about 105 to 110, usually the year 107 is given (Wikipedia offers the year 108). He was already quite old, and there is pretty unanimous agreement among those who themselves knew him, that he had known some of the Apostles in his youth. Seeing as he was a child and young man in the mid-1st century, it's hardly extraordinary to think that he did, in fact, know the Apostles.
As a "time thing" it's helpful to remember that people don't age according to centuries. A person born in the 1940's could easily have met someone famous in the 1960's and still be alive today to talk about it.
It is worth remembering, however, Christians got along fine without a Bible until the Fourth Century when the Holy Roman Catholic Church decided they needed rules to enforce.
Not how the Biblical Canon came to be.
1) there was no "Holy Roman Catholic Church" in the 4th century. In the 4th century Christians spoke of the catholic Church, where "catholic" was an adjective; and there was nothing Roman about it. The term "Roman" in "Roman Catholic" depends on who you ask: Catholics argue that Roman Catholic refers only to Catholics of the Latin or Roman Rite (as distinct from, e.g., Byzantine Catholics or Maronite Catholics who are Catholics of the Byzantine and Maronite Rites respectively); while Orthodox and Protestants tend to use "Roman" to refer to See of Rome. There was a Catholic Church in the 4th century, just as there had been a Catholic Church in the 2nd and 3rd century, but there was no "Roman Catholic Church" in the sense of a discrete ecclesiastical institution from the rest of Christendom until the Great Schism of 1054.
2) The Christian Biblical Canon was not settled in the 4th century, no more than it had been settled earlier; and debates and discussion over the canonicity of certain books continued for centuries. If you are imagining that the Bible was established at the Council of Nicea then this is incorrect, the Council of Nicea didn't discuss the Biblical Canon, it simply wasn't a topic of discussion or debate; neither (though many imagine otherwise) did Emperor Constantine play any role. It doesn't take much looking into historical source material to see that the Canon was still fluid during and after the 4th century, compare the list of canonical books offered by St. Athanasius in his 39th Festal Letter to the Canon lists offered by the local synods of Carthage and Laodicea (all 4th century), or consider that even as late as the 8th century the Apocalypse of St. John was still a highly controversial text and St. John of Damascus had to argue for its inclusion among the Eastern Churches; and even still in 1200 the Armenian Bible still did not include the Apocalypse in its Canon, though it did include the spurious text of 3 Corinthians.
In truth, there has never been a definitive closing of the Canon agreed to by all Christians. The Council of Trent settled the matter for Roman Catholics in the 16th century, various Protestant groups defined their Canon to the exclusion of the Deuterocanonical books in their confessional texts (e.g. the Articles of Religion, the Westminster Confession), there is no pan-Orthodox official closing of the Canon that I'm aware of, and there remains slight differences in one or two books between Orthodox jurisdictions. Differences of opinion between Protestants, Catholics, and Orthodox are demonstrative of the fact that the Canon of Scripture has been fluid for most of Christian history--until Trent no council, regarded as ecumenical by anyone, ever settled the matter; and only Rome considers Trent ecumenical, it is not accepted by Orthodox or Protestants.
-CryptoLutheran