"As for the Jews, who had again increased so greatly that by reason of their multitude it would have been hard without raising a tumult to bar them from the city [Rome], he [Claudius] did not drive them out, but ordered them, while continuing their traditional mode of life, not to hold meetings." (Cassius Dio Roman History 60.6.6-7)
Here we have what seems like a straightforward irreconcilable contradiction: Either Claudius' expelled the Jews or he did not. Yet it is not that simple. I recently read a historian who tried to harmonize the accounts by saying that Cassius Dio was referring to an earlier point in Claudius' reign. So there is at least one possible way to reconcile these accounts. I have not even begun to look into this to see if this harmonization actually works, but my point is that this is the normal way of doing history: considering possible harmonizations.
But more to the point: even if they do actually contradict each other, we don't throw out Suetonius and Cassius Dio and say we can't learn anything from them because they contradict each other. In fact, they are both considered as generally reliable historians.
At the places that they agree, in fact, we have much greater confidence that those are historical facts that we can rely on. This is how history is done. But skeptics refuse to concede these fundamental rules of doing history for the Gospels.
