MEXICO CITY  “Piece of cake,” the three police officers might have thought when they spied the rental car with five American tourists driving down the main drag of Cancún’s hotel zone.

They pulled the car over and told the driver, Scott Fischbach, that he was going about a mile an hour over the speed limit, recalled his wife, Michelle L. Fischbach. One of the officers cupped his hands and asked Mr. Fischbach to blow into them. “This was their Breathalyzer,” Mrs. Fischbach said, recounting her astonishment on the last night of her vacation with her husband, his sister and brother-in-law and their daughter. “They tried very hard, but Scott doesn’t drink.”

The police took his driver’s license and told him they would take him to jail unless he came up with $300, she said. The patrol car escorted the family back to the hotel, where she says the group came up with the money. The officers declined to write a receipt.