Most of the books included in the New Testament were composed in the latter half of the first century. However, none of these ancient documents have survived. All we have are copies. Furthermore, for well over one thousand years, these copies were made by hand. Scribes often made mistakes, and sometimes they even changed the text intentionally. Thus the surviving manuscripts contain many thousands of discrepancies.
When people hear this information, they often imagine that the text has been hopelessly corrupted, just like the final message in the telephone game. However, as illustrated below, the telephone game is actually a poor analogy for the transmission of the New Testament.
In the telephone game, each person in the chain passes the message on to only one other person. For this reason, a change introduced by any one person will propagate through the entire chain. Furthermore, the message is only reported by the final person in the chain. Thus the original message is lost beyond recovery.
However, as Christianity spread throughout the Roman world, New Testament manuscripts were copied multiple times. For this reason, a change introduced by any one scribe could only affect a portion of the tradition. Furthermore, while only one person reports the message in the telephone game, over 5,000 Greek manuscripts have survived. Thus, while the original documents are lost, the original text is preserved within the manuscript tradition and can be recovered by textual critics. (Textual critics are scholars who compare variant readings in order to reconstruct the oldest version of the text.)
In a modern edition of the Greek New Testament, you will find the reconstructed text on which your English translation is based. However, you will also find below the text a list of the variant readings preserved in the manuscripts. (This list of variants is called the critical apparatus.) To be sure, scholars occasionally disagree on which reading is to be preferred. The key point, however, is that the original reading has not been lost. The original reading is preserved either in the reconstructed text or in the critical apparatus.
Take for example John 1:18. Some manuscripts read "only begotten Son" (υἱός), while others read "only begotten God" (θεός). The editors of our modern critical editions favor "God," but some scholars defend "Son." Both readings are supported by early manuscripts, and the choice between the two is difficult. Nevertheless, while scholars disagree on which of these two readings is original, they do not doubt that the original reading is one of these two. In other words, the original reading of John 1:18 has been preserved, even if scholars are not sure which manuscripts preserve the original reading. By contrast, if the text had been transmitted through the "telephone game" model, we would only have the corrupted reading (represented in the image by red rectangles). The original reading (represented by blue rectangles) would be unknown to us - lost beyond recovery.