How Do I Know If Someone Is Being Honest?

Two individuals both take an oath in court.  They are asked, "Do you promise to tell the truth, the whole truth, and nothing but the truth, so help you God"?  Both say, "I do!"  Yet when the two take the witness stand, their stories contradict each other.  Both cannot be telling the truth.  One of them has to be lying, yet both seem to be sincere.  Both have good reputations.  Both speak convincingly.  Which one is telling the truth?  Tough issue, right? How can individuals lie so convincingly?