Can The Copies Be Trusted?

Not only is the New Testament reliably transmitted through the generations, but it is unique in some very interesting ways. Most events that took place in the ancient world were never written about because it was expensive to write things down and many people were illiterate. Most things that were written in the ancient world were never copied, because copying was expensive. Of the few documents that were copied, very few survived through the ages because they had to be repeatedly copied throughout the centuries. The materials that were used for copying didn’t last very long. Most of the historical documents we have from ancient times, we have a very few copies. 
Frederick Bruce gives us a sense of what would be considered historically reliable from other extant ancient documents. In his book, The New Testament Documents: Are They Reliable? he lays out the following numbers.⁠1 

Ceasar’s Gallic War: Only ten good copies survived and were made about 900 years after the events took place. 
Roman History of Livy: Only about a quarter of the original text survives, and comes from about 20 copies made around 300 years after the events took place.
Histories of Tacitus & Annals: Less than half the original text survives, which comes from 2 copies made more than 700 years after the events took place.
History of Thucydides & History of Herodotus: Only eight copies survive, which were made about 1300 years after the events took place.

The above examples are fairly indicative of historical sources. Fredrick Bruce goes on to explain that no credible historian would claim that any of the above historical documents are in question. They are considered reliable by the historical method. 
At this point we should compare the above writings with the amount of source material we have for the New Testament. It is widely known that the New Testament has been preserved in more manuscripts than any other ancient work. There are over 5800 complete or fragmented Greek Manuscripts. There are around 10,000 manuscripts in Latin, and another 9300 in various other languages.
The Chester Beatty Biblical Papyri, is a collection of very early fragments and manuscripts that have survived. The first full manuscripts of the four Gospels and Acts, were copied between 130 and 180 years after the last recorded event took place. There are fragments of copies that date to around 60 years after the last recorded events in Acts.⁠2 We even have a fragment of the Gospel of John that dates back to the end of the eye-witness period.⁠3 
The fact that we have fragments that were copied within the eyewitness period is huge. It means that we can be assured that previous copies of the New Testament were around when eyewitnesses were still able to agree or disagree with the accuracy events recorded. 
For most ancient works there are less than 35 copies. Those copies are usually made more than a 1000 years after the events took place. The new Testament has thousands of copies, the earliest of which date to around a century of the events they describe. This doesn’t mean, by itself that the events are true, only that the copies are more reliable representations of the originals than any other ancient work.

1 Bruce, FF. “The New Testament Documents: Are They Reliable.” (2003): p. 11
2 Ibid. p. 12

3 The eyewitness period is the period of time following the biblical events, where eyewitnesses of the events would still have been alive. This is important, because eyewitnesses could corroborate or deny the story.