Indeed, I recently had the pleasure of interacting with a lay apologist for Catholicism who, buying into the myth, apparently believed that people didn't have Bibles before the sixteenth century. He didn't say so, but I think he was somewhat surprised to discover just how many European and non-European languages the Bible had already been translated into before Luther ever nailed his 95 theses to Wittenburg door on October 31, 1517. He seemed to have a mental picture of the Bible surviving from the apostles to the reformers in Latin copy locked away in bishops' chambers.
But this picture of the world is far from the reality that Bibles were the treasured possession of the faithful since the time of the early church. Even in Europe, before Luther, Wycliffe and his followers produced manuscript (hand-written) English Bibles for the laity in an era and religious climate in which such production could get one killed. Others produced Bibles in the common tongue long before that, with Jerome publishing the Bible in Latin not to place out of reach of the common man, but to place it into the common tongue.
But even if someone is well-versed in the fact that Jerome produced a Bible - one may wonder whether his monumental task of producing a Latin Bible from copies of the the Greek and Hebrew originals was the first time that the Bible had been assembled as such. It was not.
Codex Vaticanus and Siniaticus represent essentially complete "codex" forms in which the whole Bible was assembled as a collection. They are normally dated to the 4th century.
Even earlier, the notable papyrus P72 (which is often dated to the 3rd century) contains what appear to be page numbers at the top, which indicate that the books contained in it were at some point bound together into a book form. It's less impressive in the case of P72, because in that case Scripture was bound together with other books that are plainly not Scripture.
Nevertheless, it is interesting to note something even more significant than the page numbers: the heading of the section of the papyrus at which 2 Peter begins reads: "Peter's Epistle 2" using the Greek letter Beta to represent the number two. That single character, Beta, has an important significance for the issue of the canon of Scripture. That single character demonstrates the fact that at least a mental and/or implicit canon of Scripture already existed in which there were (at least) two epistles of Peter.
That is page Kappa Gamma (23). On the facing page, Kappa Beta (22), we have the end of Peter's first epistle, which is similarly indicated at the foot of the epistle by "Peter's Epistle 1" using the Greek letter Alpha to represent the number one. Thus, we can see not only that the books were identified by the designations A and B, but they were even bound in that order.
We may take that kind of designation for granted, because we are used to Bibles today with their handy table of contents (canon). Nevertheless, that kind of designation demonstrates already in the earliest documents that we have, a recognition of the fact that there was more than one epistle of Peter, and that apparently as early as A.D. 200 there was already an established convention as to which epistle was A (1 Peter) and which was B (2 Peter), such that someone would use such an abbreviation in the heading of the book. P72 is interesting because, although it is no longer complete, it contained both epistles of Peter as well as the epistle of Jude, which further serves to demonstrate that the epistles did not simply circulate as individual letters in the early church.
Furthermore, P46 (dated to about 250 A.D.) contains a collection of Paul's epistles, while P45 (of similar date) contains a collection of the gospels and Acts. The only early physical copy of the New Testament is a single fragment (P52) that is from John's gospel, but which is too small to determine whether it was part of a larger codex.
In short, the physical documents we have speak to the fact of the recognition of the canon by the earliest Christians, and attempts by early Christians to bind the canonical books together into Bibles.
With that archaeological background we should not be surprised to hear Augustine testify:
Call this fancy, if it is not actually the case that men all over the world have been led, to believe in Christ by reading these books.(NPNF1, Vol. IV, Reply to Faustus the Manichaean, Book XVI, Section 20.)
Nor should our ears be shocked to hear Chrysostom telling his congregation:
Wherefore I exhort you both to obtain Bibles, and to retain together with the Bibles the sentiments they set forth, and to write them in your minds.(NPNF1, Vol. XIV, Chrysostom's Homlies on the Gospel of John, Homily 53.)
Of course, we will gladly acknowledge that Bibles in the ancient world were not as cheap and easy to get as Bibles are today. Praise be to God for the printing press and the Internet! And surely in many places today, the population is literate to a greater degree than ever before. Praise be to God for this advance in education! But recall what the solution was in the days of Caesarius, bishop of Arles (470-543):
Moreover, since what a man procures in this life by reading or good works will be food of his soul forever, let no one try to excuse himself by saying he has not learned letters at all. If those who are illiterate love God in truth, they look for learned people who can read the sacred Scriptures to them.(Fathers of the Church, Vol. 31, Sermon 8.1 of Caesarius of Arles)
And even before Christ, the Word of God was read to the people of God, even if there was not then a full scroll in every hand, Scriptures says that they read from the Scriptures:
2And the seed of Israel separated themselves from all strangers, and stood and confessed their sins, and the iniquities of their fathers. 3And they stood up in their place, and read in the book of the law of the LORD their God one fourth part of the day; and another fourth part they confessed, and worshipped the LORD their God.
For this is God's command and the way to preserve the faith:
16Take heed to yourselves, that your heart be not deceived, and ye turn aside, and serve other gods, and worship them; 17And then the LORD'S wrath be kindled against you, and he shut up the heaven, that there be no rain, and that the land yield not her fruit; and lest ye perish quickly from off the good land which the LORD giveth you. 18Therefore shall ye lay up these my words in your heart and in your soul, and bind them for a sign upon your hand, that they may be as frontlets between your eyes. 19And ye shall teach them your children, speaking of them when thou sittest in thine house, and when thou walkest by the way, when thou liest down, and when thou risest up. 20And thou shalt write them upon the door posts of thine house, and upon thy gates: 21That your days may be multiplied, and the days of your children, in the land which the LORD sware unto your fathers to give them, as the days of heaven upon the earth.