Question: My wife and I went to Detroit Metropolitan Airport with two tickets to Puerto Vallarta, Mexico, that we bought from Aeromexico. The flight was operated by Delta Air Lines. We checked our bags, received boarding passes, and went to the gate.

But when we tried to board the flight, they turned my wife away without giving us a reason. Delta rebooked us on a later flight.

At the gate for the later flight, a Delta representative turned me away. I called Delta and a customer-service representative told me that I would have to buy a new ticket for $764. I reluctantly paid this, since our bags were already on their way to Mexico and I saw no other good options.

I would like Delta to refund the $764. Can you help me?

  Answer:  Delta should have let you board the first flight. When you asked what happened, Delta blamed your denied boarding on a "system error."

"You have my sincerest apologizes  [sic] for any unfavorable impression in this instance; I certainly understand how  frustrating this situation must've been," a Delta representative wrote in an email. "I want you to know I have forwarded your  information to our Airport Operations leadership team for internal review to better our service for your future travel."

As a "goodwill gesture," Delta offered you a $75 Delta Choice gift. When you responded that you wanted the $764 as  requested, Delta offered two $50 gift certificates in addition.

This case is a little complicated. Delta acknowledges that it experienced a system error, but it also says that there was a seat for you on the first flight and that your seat flew empty. So you had the option of traveling as scheduled but decided to stay with your wife, who did not have a seat. Given that a second airline, Aeromexico, was involved, I'm not sure if anyone at either airline knows exactly what happened.

I'm tempted to say that this is a cautionary tale about code-sharing (your Aeromexico flight being operated by Delta), but I have no idea what went wrong. I also might say something about using a travel agent, but there's no telling whether using an agent could have prevented this.

Further complicating your case: You had filed a credit card dispute. A credit card dispute is the nuclear option. Once your bank sides with a company, which is exactly what yours did, your next step is small claims court.

A direct, written appeal to one of the executive contacts at Aeromexico (elliott.org/company-contacts/aeromexico) or Delta (elliott.org/company-contacts/delta-airlines) might have fixed this for you. I list the names, numbers, and email addresses of key managers on my consumer-advocacy site. You say that you did so, but that no one responded.

I contacted Delta on your behalf. The airline refunded the cost of your ticket.

  Christopher Elliott is the ombudsman for National Geographic Traveler magazine and the author of "How to Be the World's Smartest Traveler." You can read more travel tips on his blog, elliott.org, or email him at chris@elliott.org.