Good question. Even the King James Bible says on the title page "...and with the former translations diligently compared and revised." No one who really knows the Bible (and who knows people) thinks that there were no errors in copying the Bible. And they don't need to. Because there is no other book in the world that was hand copied so carefully, in which the mistakes were so carefully catalogued and documented, and certainly no other ancient book that was spread by means of so many copies . Oddly enough, this PROTECTED the original Bible's integrity, so that in our day it is possible to have a Bible with practically no mistakes, certainly no serious ones.
How? Think about it. Lots of different people making copies of the Bible at all periods of history. Lots of very human people making an occasional human mistake. But NO TWO PEOPLE MAKING A MISTAKE IN EXACTLY THE SAME PLACE. So if you got copies of the Bible together from many different people, and went through the Bible verse by verse, ALL COPIES WOULD SAY THE SAME THING IN EVERY VERSE EXCEPT THE ONE WITH A MISTAKE. This would make the mistake stand out like a sore thumb, so it could then be corrected. Do the same with the next verse, and on through the entire Bible. This science is called "textual criticism" and has resulted in it being possible to produce a Bible now with nearly no mistakes.
Thus, when the Dead Sea Scroll of the Bible book of Isaiah was discovered, ELEVEN HUNDRED YEARS OLDER than the oldest scroll that they had at the time, it was found to say exactly the same thing, with only minor difference of punctuation and spelling even after eleven hundred years of copying. Thus, Sir Frederick Kenyon, a director of the British Museum, wrote fifty years ago, 'Thus, the final basis for doubt that the Scriptures have come down to us substantially as they were originally written has now been removed.' This hasn't stopped many religions from translating and interpreting it to suit themselves, though, but the Bible actually said that that would happen (Matthew 13, the "weeds among the wheat") forcing sincere persons to "dig" to get at the truth, which no sincere person minds doing. And you'll have to do, too, if you want the truth. So the real question is, how bad do you want the truth? (compare John 4:23,24))
"All Scripture Is Inspired of God and Beneficial", Watchtower Bible and Tract Society of Pa.(Jehovah's witnesses)