Various kinds of difficulties are alleged:-

1. Contradictions between different accounts, e.g. was the ascension immediately after the resurrection (Luke 24), or 40 days later (Acts 1)?
2. Contradictions between the biblical account and other sources, e.g. the census of Quirinius (Luke 2:1) took place well after the time of Herod.
3. Insufficient attestation from external sources, e.g. the alleged custom of releasing a prisoner at Passover time (Mark 16:6; Luke 23:25) is not attested elsewhere.
4. The scriptural accounts relate unlikely or impossible events, e.g. the number of soldiers accompanying Paul seems improbably high, Acts 23:23.
5. It is not possible to see how the author could have known about what he was relating, e.g. in Acts 26:30-32 Luke records what happened behind closed doors.
6. It sometimes seems easier to view biblical narratives in literary, rather than historical terms, e.g. various features of the Pentecost story, Acts 2-3.

Of course, some of the difficulties only affect details of the narrative and do not call into question the historicity of the account as a whole.  Others, however, raise more serious doubts.  In either case, questions are raised about the nature and extent of inspiration.

Would the established existence of even a single, incontrovertible error be sufficient to cast doubt on the doctrine of biblical inspiration?

Historical study is always somewhat provisional: it is based on probabilities rather than proof and is open to new evidence or new interpretations of the evidence.  It follows that the historical infallibility of Scripture cannot be proved absolutely.  It is the task of apologetics to establish the level of probability with regard to the historicity of the narratives.

With regard to supernatural events, there are those that seem by ordinary standards incredible (such as miraculous healing) and case where the cause of an event is said to be divine (such as the death of Herod, Acts 12:23).  We need to consider (a) whether our world view includes the possibility of miracles; (b) if so, whether the miraculous explanation is the best in a given instance; (c) whether we can agree that even a ‘natural’ phenomenon can have a ‘religious’ explanation.

There is the broader question of the intentionality of the writers.  However, in the case of Acts, for example, a strong case can be made that Luke purports to be writing history.

It is important to consider what would count as an historical ‘error’.  Inexactness is not necessarily error, and neither is paraphrasing when reporting the words of another.

Some approaches to harmonisation can stretch credulity, as when the accounts of Peter’s denial are added together to make a total of six denials.  In Acts, the three accounts of Paul’s conversion vary quite considerably, but the fact that they are all the product of one author shows that he did not regard them has discrepant.

Where apparent discrepancies are found between biblical and extra-biblical authors, we should not always privilege the latter.  Josephus, for example, had his own biases, misinformation and carelessness.  Scholars from W.M. Ramsay to C.J. Hemer have gathered evidence that tends to confirm the biblical narratives.

In the case of the Gospels, a particular problem is the way in which some scholars seem to assume that if a theological motive can be found for an Evangelist recording a saying or action of Jesus in a certain way, this is sufficient to say that the writer invented it.

To return to the examples at the beginning:-

1. Contradictions between different accounts, e.g. was the ascension immediately after the resurrection (Luke 24), or 40 days later (Acts 1)? – Luke might have condensed his narrative on the first occasion and expanded it on the other.
2. Contradictions between the biblical account and other sources, e.g. the census of Quirinius (Luke 2:1) took place well after the time of Herod. – Fresh evidence may indicate that we should translate this, ‘the census before Quirinius was governor’).
3. Insufficient attestation from external sources, e.g. the alleged custom of releasing a prisoner at Passover time (Mark 16:6; Luke 23:25) is not attested elsewhere. – Opinions differ as to whether a regulation in the Mishnah reflects this.
4. The scriptural accounts relate unlikely or impossible events, e.g. the number of soldiers accompanying Paul seems improbably high, Acts 23:23.  – A force of 300 men in not excessive if an ambush of 40 terrorists was expected.
5. It is not possible to see how the author could have known about what he was relating, e.g. in Acts 26:30-32 Luke records what happened behind closed doors. – Luke was perfectly capable of reconstructing the gist of what was said.
6. It sometimes seems easier to view biblical narratives in literary, rather than historical terms, e.g. various features of the Pentecost story, Acts 2-3. – It is not impossible that there are symbolic elements (e.g. tongues of fire) in a narrative that is fundamentally historical.

As Calvin said, the Holy Spirit was not too worried about trifling matters.

Based on article by I.H. Marshall in New Dictionary of Christian Apologetics.