A tangle of factors make diagnosing head injuries in the military particularly tricky, experts say. Some troops try to hide symptoms so they can stay on duty, or avoid being perceived as weak. Others may play up or even invent symptoms that can make them eligible for the Purple Heart medal or valuable veteran’s education and medical benefits.

And sometimes commanders suspect troops with legitimate injuries of malingering and force them to return to duty. Pentagon officials said privately this week that some of the injuries from the Jan. 8 incident had probably been exaggerated. Mr. Trump seemed to dismiss the injuries at a news conference in Davos, Switzerland, last month. “I heard they had headaches,” he said. “I don’t consider them very serious injuries relative to other injuries I have seen.”

In the early years of the war in Iraq, troops with concussions were often given little medical treatment and were not eligible for the Purple Heart. It was only after clearly wounded troops began complaining of poor treatment that Congress got involved and military leaders began pressing for better diagnostic technology.

Damir Janigro, who directed cerebrovascular research at the Cleveland Clinic for more than a decade, said relying on the questionnaire makes accurate diagnosing extremely difficult.

“You have the problem of the cheaters, and the problem of the ones who don’t want to be counted,” he said. “But you have a third problem, which is that even if people are being completely honest, you still don’t know who is really injured.”

In civilian emergency rooms, the uncertainty leads doctors to approve unnecessary CT scans, which can detect bleeding and other damage to the brain, but are expensive and expose patients to radiation. At the same time doctors miss other patients who may need care. In a war zone, bad calls can endanger lives, as troops are either needlessly airlifted or kept in the field when they cannot think straight.

Mr. Janigro is at work on a possible solution. He and his team have developed a test that uses proteins found in a patient’s saliva to diagnose brain injuries. Other groups are developing a blood test.