There are three different claims bundled into that question — that copyists made errors, that translators distorted the meaning, and that someone deliberately rewrote the text. The first is true and thoroughly documented; the second is manageable; the third has no evidence and, more importantly, no opportunity.

1. Three claims, three different answers. Collapsing them is what makes the objection feel stronger than it is. Copying errors are a fact we can measure; translation is a skill we can check against the original languages; deliberate doctrinal rewriting is a historical conspiracy claim, and it can be tested like any other. Note too what the Bible actually claims for itself — not that every copy was perfect, but that the originals came from God through men: "no prophecy was ever produced by the will of man, but men spoke from God as they were carried along by the Holy Spirit" (2 Peter 1:21).

2. The manuscripts are embarrassingly abundant. We have more than 5,800 Greek manuscripts of the New Testament, and roughly 24,000 when Latin, Syriac, Coptic, and other versions are counted. Homer's Iliad survives in about 1,900 manuscripts, the earliest roughly a thousand years after composition; Caesar's Gallic Wars in around ten, with a similar gap. No ancient document is remotely in the New Testament's league. Abundance is what makes reconstruction possible — with one copy, you cannot tell a copying error from the original.

3. The gap is decades, not centuries. P52, a fragment of John, is dated to about AD 125, within a generation of the original. P46, P66, and P75 date to around AD 200. Codex Sinaiticus and Codex Vaticanus, both fourth century, contain the entire Bible. For most classical authors the earliest surviving copy postdates the original by five hundred to a thousand years. Here it is decades — which means the text was in wide circulation while people who had heard the apostles were still alive.

4. Variants exist, and we publish them. The differences number in the hundreds of thousands, and the majority are trivial: spelling, word order, a stray definite article. Bruce Metzger and Daniel Wallace both judge that no variant affects a doctrine of the faith. Be fair to the other side, though: Bart Ehrman disputes that slogan, and two passages are genuinely disputed — the long ending of Mark (16:9-20) and the woman caught in adultery (John 7:53-8:11). Modern Bibles bracket them precisely because the evidence is contested. You can see the argument in the footnotes.

5. The church fathers are a second witness. The ante-Nicene writers — Clement, Ignatius, Justin, Irenaeus, Tertullian, Origen — quote the New Testament so heavily that large stretches could be reconstructed from their works alone, and they were reading manuscripts we no longer possess. That is an independent check on the copying process, from a different century and different cities.

6. Translation is not corruption. Every serious translation is made by a committee from the Greek and Hebrew, and the differences between faithful translations are visible and small. Where they differ — usually on how to render a single word — you can look it up. A paraphrase is a different product, and it says so on the cover.

7. Turn: no center could have coordinated it. To rewrite the text everywhere, someone would have to control every copying center. By AD 200 the New Testament existed in Greek, Latin, Syriac, and Coptic, across three continents, with no pope, no printing press, and no way to coordinate. And the motive fails on its own terms: a church that wanted to invent a flattering text kept Peter's denial, the disciples' cowardice, Paul's confession of persecution, and women as the first resurrection witnesses. Editors with that much power do not leave that material in — and a church that believed Scripture is "breathed out by God" (2 Timothy 3:16) had reason to copy carefully, not to improvise.

8. This is the most checkable claim in the Bible. You do not have to take any of this on trust. The critical apparatus in the Nestle-Aland Greek New Testament lists the manuscripts behind every disputed reading, and the manuscripts themselves are photographed and published. If the text had been systematically changed, the evidence of the change would have to be in the copies — and the copies are the thing we have most of.

The honest summary is that the New Testament has been copied by fallible people, that we know about the errors because we have so many copies, and that the text we have is reconstructed from the evidence rather than asserted. Christians have always held that God's word stands forever (Isaiah 40:8) and that Christ's words will not pass away (Matthew 24:35); the claim here is narrower and checkable — the reconstruction is public, and you can read it.

Sources: Bruce Metzger, The Text of the New Testament; Daniel B. Wallace; Michael J. Kruger, Canon Revisited; Bart Ehrman (for the skeptical case); Philip Comfort, Encountering the Manuscripts.