The pilots began noticing the objects after their 1980s-era radar was upgraded to a more advanced system. As one fighter jet after another got the new radar, pilots began picking up the objects, but ignoring what they thought were false radar tracks.

“People have seen strange stuff in military aircraft for decades,” Lieutenant Graves said. “We’re doing this very complex mission, to go from 30,000 feet, diving down. It would be a pretty big deal to have something up there.”

But he said the objects persisted, showing up at 30,000 feet, 20,000 feet, even sea level. They could accelerate, slow down and then hit hypersonic speeds.

Lieutenant Accoin said he interacted twice with the objects. The first time, after picking up the object on his radar, he set his plane to merge with it, flying 1,000 feet below it. He said he should have been able to see it with his helmet camera, but could not, even though his radar told him it was there.

A few days later, Lieutenant Accoin said a training missile on his jet locked on the object and his infrared camera picked it up as well. “I knew I had it, I knew it was not a false hit,” he said. But still, “I could not pick it up visually.”

At this point the pilots said they speculated that the objects were part of some classified and extremely advanced drone program.

But then pilots began seeing the objects. In late 2014, Lieutenant Graves said he was back at base in Virginia Beach when he encountered a squadron mate just back from a mission “with a look of shock on his face.”

He said he was stunned to hear the pilot’s words. “I almost hit one of those things,” the pilot told Lieutenant Graves.

What was strange, the pilots said, was that the video showed objects accelerating to hypersonic speed, making sudden stops and instantaneous turns — something beyond the physical limits of a human crew.

“Speed doesn’t kill you,” Lieutenant Graves said. “Stopping does. Or acceleration.”

Asked what they thought the objects were, the pilots refused to speculate.