Dear Tripped Up,

My husband and I had flown nine times without incident on the same route (Seattle to Houston) with the same dog (Bella, our 13-year-old Coton de Tulear) in the same carrier (soft-sided, correct dimensions) on the same airline (United). But on trip number 10, the counter agent in Seattle told us that because Bella’s back touched the top and sides of the carrier when she turned around, she could not fly. A supervisor backed up the decision. We walked down the concourse to Delta, bought expensive one-way tickets, and flew the next day without incident. I was shocked that an airline can arbitrarily enforce a rule that appears nowhere on its website. Can you get to the bottom of this and help us get compensated for the premium we had to pay for a last-minute flight? Jacki, Seattle

Dear Jacki,

You’re not the only one to complain to me about a mysterious “no touching” rule that, though unwritten, seems to have gained traction in the oral tradition of airline check-in agents. Betsy of Paradise Valley, Ariz., wrote in with a very similar story on American Airlines, when her family’s two Havanese were initially denied boarding because they could not “stand fully erect without their head touching the top of the crate,” she wrote. Though the airline eventually gave in when shown documentation that the dogs had been allowed to fly the day before, Betsy told me her family has not dared to fly with their dogs since, and instead have been driving cross-country.

A glance through the airline reviews on BringFido, a pet travel site, found several more “no touching” incidents that resulted in a dreaded one bone review (out of a potential five).

Traveling with pets has been getting harder in recent years — emotional support animals are out, fewer airlines allow pets to travel in the cargo hold, and the very small dogs, cats and occasional birds and other pets that are allowed to fly in-cabin have to follow very strict rules that vary, almost maddeningly, by airline. The U.S. government, though very concerned about what dogs and cats enter the country, has taken a hands-off approach to domestic pet travel, leaving the individual airlines to set their own specific guidelines as long as the animals are in containers that fit fully under the seat in front of the passenger.

This means each airline has its own rules about what animals can fly in the cabin, usually for a fee of $95 to $120, and the specific dimensions of the carriers (which can even vary by aircraft). Furthermore, some prohibit certain breeds of dogs, where others allow any dog, cat, household bird or other critter that fits weight restrictions, most commonly a 20-pound-plus-carrier weight limitation that is strictly enforced and can lead to last minute treat-restriction. Spirit has a 40-pound limit and allows household birds, raising some intriguing questions about pet turkeys I don’t have time to get into here.

But the language on how to tell whether a pet fits into its carrier is generally the same, probably because it’s the one area where both the government and the International Air Transport Association, or IATA, an international trade association, have weighed in. Federal regulations state animals must have “enough space to turn about normally while standing, to stand and sit erect, and to lie in a natural position,” and IATA’s guidelines are quite similar.

I contacted both United, for your case, and American, for Betsy’s, and got two completely different answers.

United requires that pets be able to “stand up and turn around while inside.” But what that means is subject to some interpretation. “In order to be accepted,” Erin Jankowski, a United spokeswoman, wrote to me in an email, “a pet must be able to stand up and turn around without rocking or tipping over the carrier.”

I followed up, sending along your complaint in full. The answer was short on specifics, but direct. “In this particular case,” she responded, “the agent made the correct decision based on our guidelines.” She added that you received a refund within days for the original cost of your flight, but did not address why Bella had been allowed on previous flights.

American Airlines had a different take for Betsy and her family’s Havanese. “It’s permissible for the pet’s fur, tail or ears to touch the inside of the kennel or carrier,” wrote Andrea Koos, an American spokeswoman in an email. “They can fly with their dogs, based on the information you provided.”

I wrote to several other major American carriers, and of those that responded (Delta, JetBlue, Alaska and Frontier), only Delta cited a “no touching” rule. In other words, though airline employees do have discretion on deciding whether a container is humane and whether it will fit under the seat, a pet on most airlines should not be turned down solely because its fur touched the side or top of a carrier.

But depending on how much it touched — especially if it caused a bulge — that could be reason enough to reject a pet. Sheila Goffe, a vice president of the American Kennel Club, noted that dogs with erect ears that add a couple inches to their height and pressed up against the top of the container could, in theory, be turned away. “It’s a funny thing to have to think about, but you get into gray areas that could be a problem,” she said.

Here’s the bottom line, according to Erin Ballinger, the destinations editor for BringFido: “Travelers are always going to be at the mercy of the airlines’ discretion.” If it’s a close call as to whether your pet has enough room, you may want to rethink your travel plans.

“The very obvious solution” for borderline cases, said Dr. Nelva Bryant, founder of the site When Pets Fly, “would be to send the pet as cargo and not risk getting denied.” Of course, many dog owners dread the idea of putting their pet in the cargo hold, and many airlines (including Delta and United) have stopped flying animals as cargo, at least for now.

Another solution — though not one I’m assuming most dog owners will go for — would be to get a cat. But cats aren’t completely immune to problems, as I found out when Frederick from Los Angeles wrote to tell me about his domestic shorthair tabby named Lola, who was denied boarding on a Delta flight last December, even though she easily fit into a regulation carrier.

It turns out that pets, like humans, can be bumped.

Airlines allow a maximum number of pets per plane and even per cabin within the plane (service animals are exempt from limits) and strongly recommend reserving in advance. Delta, like several other airlines, has an explicit “first-come, first-served” policy posted for those who don’t. But Frederick made a reservation, so why were he and Lola bumped and forced to take a later flight? Could Delta have overbooked pets on the flight?

According to Morgan Durant, a spokesman for Delta, that’s not what happened. “Those who call before the departure date get served before anyone else would” as long as they were there two hours before departure, he wrote in an email.

Frederick said he did get there two hours ahead of time, but then waited in what he described as a “hideously slow line” past the two-hour mark. I presume that Delta gave away Lola’s spot to a walk-in dog or cat (or fly-in parakeet) in line ahead of them. That was clearly not his fault, and Mr. Durant told me Delta had apologized for the error and made a “good will gesture” to Frederick.

Frederick said the only thing he got was an upgrade to Comfort Plus on his flight four and a half hours later, and only when he asked. He said that wasn’t enough, and I’d tend to agree: When someone is traveling with a cat, you can’t just throw them a bone.