I blogged on this a little while back, and thought it was time to explore a couple of questions a bit more. After Easter the set Bible readings for the Church of England follow the account of Jesus rising from the dead and appearing to his disciples. Quite apart from the step of faith required to believe that at all, this has a few problems.
Mark's Gospel (usually seen as the earliest) ends abruptly at Mark 16:8 in the earliest Greek manuscripts available: "and they said nothing to anyone, for they were afraid". Not exactly the ending you would want at the climax of the greatest story ever told! Later manuscripts have verses 9-20 or another ending that include other appearances and summarize stories found in the other Gospels. The interesting thing is that some evangelical Christians quote the later verses as if they had the same authority as the rest of the Gospel. But, of course, if you have an inerrancy/infallibility view of Scripture, this passage can't be included. On the best evidence available, it isn't Scripture "as originally given"; it is New Testament apocrypha
Or take John 5:1-9, where a man is healed at the pool of Bethesda/Bethzatha. The earliest Greek copies omit verse 4, which refers to the story that when the pool water is disturbed, it is an angel doing it, and that is the moment to enter the water for healing. What seems clear is that verse 4 is a footnote added later to clarify what the man says in verse 7. In later versions, it got incorporated into the main text, although most modern translations make it a footnote. For people with very scrupulous views about Scripture, the status of this verse becomes problematic.
There are plenty more. The one I was studying recently is the fact that the story of the Ethiopian eunuch in Acts 8 didn't originally have Acts 8:37. The New Revised Standard Version footnote states Other ancient authorities add all or most of this, [And Philip said, "If you believe with all your heart, you may." And he replied, "I believe that Jesus Christ is the Son of God."
The original story simply has the eunuch saying what is to prevent him from being baptized, and Philip does it. The extra later verse inserts a credal statement that presumably reflects the practice of the early Christians. Someone obviously wanted baptism to be seen to require some greater understanding than the original text implies.
The boundaries of the 'canon' of Scripture are, in reality, a bit more messy than we often like to portray, and so it seems to me that an excessive emphasis on inerrancy of the Bible runs into problems the moment you start taking the text itself seriously.