This past weekend (12/14 & 15/'13) I taught Acts 5:12-42, where Peter and the other apostles obey God's command to proclaim the Good News about Christ, and in so doing disobey the Jewish authorities to shut up about Jesus. The clash of authorities isn't resolved in the passage, but each side is determined not to back down.
Since I'm a visual learner, I devised this chart to better help me understand and communicate what to do when two legitimate authorities say opposite things: "When any authority commands you to do what God forbids, or forbids you to do what God commands, you must obey God."
In all things, then, we are subject to authority––we recognize it, respect it and respond to it (Romans 13:1, 1Peter 2:13-15). But if any authority (whether government, parents, employer, husband, or church) clashes with God's, and commands you to do what He forbids, or forbids you do what He commands, always obey God, the highest authority.