We have about 5800 New Testament Greek manuscripts. However, if everyone of these were suddenly obliterated, we could reconstruct the New Testament from the early versions with considerable confidence. For example, we have about 15,000 Latin manuscripts.
I myself specialize in Coptic manuscripts. If I only had access to the New Testament in Coptic, none of my theology would change an iota, and none of the story of Jesus would change either.
I sat in on the Classics and Divinity combined PhD seminar at Cambridge one day. A paper was being presented on the neo-Platonic philosophers. They were discussing the nature of the soul as taught by these philosophers of late antiquity. I could hardly refrain from laughter as the other students made confident assertions about the beliefs held by third century philosophers such as Plotinus and Porphyry, as if the manuscript tradition were indisputable and rock solid. In reality, we have so few manuscript copies of their works (perhaps a handful)--and the few that we have are separated from them by a 1000 years and more, and perhaps only available in a secondary translation. The New Testament manuscript tradition is incredibly rich by comparison.
I don't want to exaggerate the strength of the New Testament manuscript tradition. For example, only about 15% of Matthew's Gospel is attested by manuscripts of the second and third centuries. But even so, we have good evidence of a strict line of transmission from the fourth century to the second century. My own PhD thesis strengthens this conclusion significantly.
There have been some influential works written lately which undermines the reliability of the New Testament textual transmission. These books are being assigned in colleges, and many college students are being led to the conclusion that we can't trust the Bible because we don't have the original manuscripts. Pastors are often woefully inadequate in responding to our college students, and I'd like to do what I can to support pastors and our college students facing such skeptics in the university.