Imagine you're living in Judea around the beginning of the first century. You live under the rule of the big bad Roman Empire. Who are the people you'd like to avoid? Romans. Especially Romans with power. Getting dragged in front of an official of the Empire did not always end well. Just ask Jesus.
Acts of the Apostles is sprinkled with all kinds of run-ins with honest-to-goodness Roman officials. Think of them as ancient cameos:
As you can see (surprise, surprise) most of these guys don't look so good from a Christian point of view. Rounding up apostles? Trying to score political points? Looking for bribes? Beheading Christians? Shmoop is pretty sure they wouldn't be too tickled about being remembered 2,000 years later for this stuff.
But there are some cases where the Roman officials come off as okay human beings. Gallio totally swats down the charges against Paul and releases him even though it drives the Jewish naysayers in the area crazy (18:12-17). Festus, King Agrippa, and Bernice also agree that Paul has done "nothing to deserve death or imprisonment [and that he] could have been set free if he had not appealed to the emperor" (26:31-32). Bummer.
But instead of doing something to help a disciple out, these guys just wash their hands of it and walk along on their way. Is Paul going to his death? Probably. But there's a nice buffet back at the forum and who wants to miss that? Unlike the Jewish leaders who oppose Christians, they're not being purposely cruel. They're just super indifferent.
These multiple run-ins with Roman higher ups play an important part in the story.
First, whenever a disciple is arrested and dragged into trial, it's a great chance for him to go ahead and tell the whole Christian story all over again. Luke really loves him some sermons during trial. Of course, none of these Roman officials are ever converted (even King Agrippa jokes to Paul, "Are you so quickly persuading me to become a Christian?" [26:28]). If nothing else, it's good reading material.
You also have a bunch of Roman officials basically brushing off the threat Christians pose to the Empire. In fact, it's mainly the Roman officials who are Jewish that oppose the disciples most violently (like King Herod). Your average Roman doesn't seem to think Christians are anything to worry about. Luke would like everyone to take note.
Finally, all these brushes with Rome are a way to draw even more parallels between Jesus and his disciples. Especially Paul, who is declared innocent by several Roman officials and then shipped off to (probably) die anyhow. You don't get more Christ-like than that.