I would imagine that there are close to 6000 ancient Greek manuscripts of the NT. The going number was around 5000, but I do imagine that archeology has uncovered even more within the last 20 years.
But the thing is, the manuscripts of the Bible and textual criticism is not a fundementalist Christian thing only, it envolves Catholics just as much as any Protestant. It is a very exciting aspect of Biblical studies, something that I had loved as a Baptist, and something that assisted me to find the Catholic Church, and something that I apply to my studies now. Whereas Protestants seem to apply textual criticism in some way to find Biblical authority through the age of manuscripts, while some combat it in order to try to make the King James Version their authority, but Catholics can apply textual criticism to re-enforce the authenticity of our already deep and rich Catholic faith because there is nothing that poses a threat to it, as it does in Protestantism. Protestants sometimes use it to assist creating new denominations, Catholics can use it to understand our Bible even more.
As far as all 6000 being close to the original, well, that has to be clarified a little. Many of the Greek manuscripts range from the 3rd century to the 16th century, and there are some papyrus that take you to the 2nd century but they are often fragments only. But as for the texts of the manuscripts, they are very close and do not change anything doctrinally, usually they vary in spelling, or omit portions.
Also, one thing that should not be ignored is the Latin, Aramaic, and other very important Biblical manuscripts. Some of these manuscripts are older and better than some of the Greek manuscripts. One example is the gospel of Matthew which was written in Aramaic by Matthew, and we have the Aramaic NT, which carries alot of weight, and is often misunderstood as being completely a translation from the Greek, and that is not always the case.
