Here is an interesting story regarding the authenticity and accuracy of Luke's account in Acts. It is from William Mitchell Ramsay's book on Paul's missionary journeys.
The following account about Ramsay is from Logos website:
William Mitchell Ramsay is perhaps one of the most fascinating biblical scholars from the turn of the twentieth century, and his writings are full of knowledge and insight that can only come from one who has extensively experienced first-hand the archaeology and people of Asia Minor. Perhaps most well-known for his archaeological endeavors, he traveled extensively throughout Asia Minor, studying the missionary journeys of Paul and conducting archaeological research, writing numerous books on the findings and adventures of his studies, including St. Paul the Traveller and Roman Citizen. His original intent in his studies was to disprove Christianity through archaeology, but through his research he realized that the Bible was accurate and converted to Christianity.
The following account is from James M. Boice's commentary on Acts
"Acts 14:6 was an important verse in the life of Sir William Ramsay, whom I have mentioned several times in this book. Together with a few other verses, verse 6 produced a change in his thinking that brought him to a strong trust in the reliability of Scripture.
Ramsay was a classical scholar, somewhat like Heinrich Schliemann, who discovered the ancient city of Troy. He came from Scotland, and because classical scholars liked to visit the countries they were studying, Ramsay, who was studying Acts, set out for Asia Minor, what is now Turkey. Nobody knew much about Turkey in those days. Travel was difficult. Many of the ancient sites, which particularly interested Ramsay, had been lost for centuries.
Ramsay began his research, and one of the things he investigated was the boundary line between the ancient Roman territories of Pisidia and Lycaonia that seemed according to an ancient boundary marker to have been between the cities of Lystra and Derbe. That could have been an incidental and somewhat unimportant matter in itself. Boundaries can be anywhere at all. Why should it matter? But there was a puzzle in the case of this boundary in respect to what Luke had written in Acts. When Luke wrote Acts 14, he said that the apostles left Iconium and fled to “the Lycaonian cities of Lystra and Derbe,” thereby putting Lystra and Derbe in the same province. In other words, Luke differed from the apparent evidence and was therefore assumed to be wrong.
Ramsay had been brought up on the liberalism of the nineteenth century. He did not doubt that Luke had made a mistake. He was retracing Paul’s steps, studying the cities he visited and the roads he walked, trying to understand not only where Paul went but also why he went where he did. When Ramsay got to Lystra and Derbe, he discovered that the ancient boundary stone between the two cities suggested they had been in different provinces. But he also discovered that the stone had been moved. It wasn’t where it had been originally. He began to investigate the matter more carefully.
Today, if you read his book St. Paul the Traveler and the Roman Citizen and get to his account of Paul’s ministry in these cities, you will find him pointing out that once again Luke is remarkably accurate. This is because Ramsay discovered Lystra and Derbe were in the same province, the province of Lycaonia, between the years A.D. 37 and 72, but not before those dates and not afterward. That is, they were in the same province in the very years Paul was there, as Luke accurately reports.
We find situations like this frequently in serious studies of the Bible. If you want to seem very wise and popular today, you can gain attention by making a career of criticizing the Bible. Show all the places where modern scholarship “proves” that it is wrong—if you are not afraid of looking very foolish about thirty years from now, and perhaps much sooner, when the explanation of the apparent difficulty is found. However, if you want to look wise in the future, though you may be thought foolish now, you should take your stand on the integrity and complete accuracy of this Book. If you do, you will find the same sort of things Ramsay and others discovered."