The Matthew and Luke genealogies give the ancestry of man from Adam right to Jesus. Do you believe those accounts?
I believe that Matthew and Luke pulled those genealogies from Genesis and the histories (the Torah, Judges, 1 & 2 Samuel, Ezra, Nehemiah, etc.).
No one believes they are exactly accurate. Both left out generations that are mentioned even in the Old Testament.
I believe that as you trace the lineage back, there comes a point where those histories are inaccurate. I don't believe God inspired perfectly accurate history nor perfectly accurate science. A perfect book is not what God was after. He wants a people that walks by the Word of God that comes each and every day, not just the Word of God that came 2,000 years ago.
I do believe that the names are based on real stories and records handed down. There are, however, gaps in those genealogies. Everyone acknowledges that Matthew's genealogy has many gaps. His statement that there are 42 generations from Abraham to Jesus is just not true, at least not according to the Old Testament records. We can suggest that he left out generations on purpose to make some point, a possibility I would allow, but we cannot believe that God was trying to preserve a literal, accurate record because it's just not accurate.
As an example, Matthew lists Uzziah (or Ozias in the KJV) as the son of Jehoram, the son of Jehoshophat, but this skips Ahaziah, Joash, and Amaziah. That's 3 entire generations, and according to a timeline I checked (made by Christians), it's a 50-year period that he skips.
We have to acknowledge the possibility that Matthew didn't skip those on purpose. It's entirely possible that he confused Ahaziah, the real son of Jehoram, with Azariah, which was Uzziah's long name.
I know such thoughts are horrifying to Christians who have based their faith on a belief that the Bible is some sort of magic book. Originally, the Christian faith was based on no such thing. Yes, they honored the Law of Moses the same way that the Jews did, but the faith of Jesus Christ was based on the testimony of the apostles to the resurrection and the gift of the Spirit of God that was given to those who believed.
For example, when Paul said he was not ashamed of the Gospel of Christ (Rom. 1:16), he gave a reason for that. His reason was that "in it, the righteousness of God is revealed from faith to faith" (Rom. 1:17). He then goes on to point out that under the Law the Jews were teaching one thing but living another. Not so with the Gospel of Christ. The Gospel of Christ was revealing the righteousness of God in those who believed, so that they were dying to sin and living to God.
It is foolish of us to dishonestly defend a "divine accuracy" in things that are obviously not accurate. The Matthew and Luke genealogies are an example. If you're a believer in God like I am, then surely you must believe that God knows about the gap between Jehoram and Uzzia in Matthew's genealogy! Yes, he could have inspired Matthew not to leave that gap and claim 14 generations when there were really at least 17.
But he didn't!
Why should that prove God make a mistake? Why shouldn't it prove that God doesn't care about providing a miraculously accurate science and history book?
The Kingdom of God, according to the apostle Paul, does not consist of words, but of power. The New Covenant is a covenant of the Spirit, not of the letter.
When we realize that, and when we quit trying to defend the letter, maybe we will also reveal the righteousness of God from faith to faith, and that righteousness and power will leave the mockers and scoffers speechless as they behold the power and love of the kingdom of God.
I've seen it happen. I still see it happen.
I will add one more thing. Jesus once said that we can't be his disciples without—in so many words—becoming whole-hearted, single-minded followers of him. You can see the words he did use, which are much stronger than my words, in Luke 14:26-33.
Maybe if we're going to demand literal accuracy in the words of the Bible, we could start by believing Jesus' words. Maybe the reason Christians are fighting powerlessly in defense of the letter is because they don't—and won't—believe what Jesus said! Thus, most wind up without the righteousness, power, and Spirit that were the proof of Paul's Gospel.
One final word, for those who have heard that there is a conflict between Paul's Gospel and Christ's, it's not true. Historically, it doesn't even make any sense. Historically, it's an impossibility. Christians simply don't understand Paul's Gospel, and they ignore much of what he said. (For example, his emphasis on works in passages like Rom. 2:5-8; Gal. 6:7-10; Eph. 5:3-8; etc.)
I have a thorough treatise on Paul's Gospel and its relationship to the teaching of Jesus and James at my Christian History for Everyman site at http://www.christian-history.org/sola-fide.html.