Representative Filemon B. Vela Jr., a Democrat who represents Brownsville in Congress and whose district includes Los Indios, said the little piece of fence illustrated the pointlessness of a border wall, regardless of which administration built it.

“I’ve voted against every single piece of border wall funding that’s ever come up, and I’m going to continue to do so,” Mr. Vela said. “Decisions are being made in Washington in terms of where to put fencing that don’t make any sense.”

As the president suggested, this strange little spot’s days appear to be numbered. A Border Patrol official who oversees the Los Indios area, Henry Leo, the patrol agent in charge of the agency’s station in Harlingen, Tex., said plans are in the works to finally install the necessary gates in the next one to three years.

“We did request additional fencing and gates in that area, so what you see right now is not the complete picture,” Mr. Leo said. “The plan is to connect to that piece of fencing and make it a continuous fence with gates.”

Both Mr. Leo and the mayor of Los Indios, Rick Cavazos, defended the fence in the area, including the stand-alone section, saying that overall the fencing had helped decrease illegal crossings. “I saw a dramatic decline in crossings and apprehensions as a result of the barrier,” said Mr. Cavazos, who is a retired Border Patrol agent. “In 2004, 2005, 2006, this was a high crossing area. People would come up from the river. The numbers went down.”

On a hot April afternoon, it was all quiet at the fence island. The loudest sound was the soil: the dry brush, dirt and grass crunches underfoot. The pillars of the fence are as rough as sandpaper, and leave little red flakes on your fingertips that resemble chili powder.

Adding to the head-scratching quality of this fence is its location.

On one side of the fence is America. But the other side is America, too. The fence runs inland, far from the river, the middle of which is the official boundary between Mexico and the United States. That means this 36-beam fence in one sense divides the country itself. In that sense, the openings in the fence are necessary: They allow property owners and ranchers to access their land that is north of the river but south of the fence, a region locals call a “no man’s land.”