At this point we hit an unexpected obstacle: credibility. Several of us went to brief the commander of the 4th Infantry Division, Maj. Gen. William Peers. I warned him that a multi-division North Vietnamese attack on Dak To would take place at any moment, coincident with attacks throughout the highlands. He shook his head and pointed to our camp on Engineer Hill. “So I’m supposed to believe that some kind of magic allows a bunch of shaky girbs” — acronym for “G.I. rat bastards” — “distinguished more for their spit than their polish and abetted by a civilian, to use a tangle of antennas and funny talk to divine the combat plans of the enemy?” He waved us away. The briefing was over.

But we were right. On Nov. 1, a bomb dropped from a B-57 struck somewhere near Dak To. It hit an enemy supply dump, setting off secondary explosions, proof positive that significant numbers of North Vietnamese troops were out there. General Peers sent a unit from the division’s 1st Brigade to investigate and make contact with the Special Forces camp at Dak To. Two days later, one of the brigade’s battalions landed by helicopter on Hill 978, near Dak To, expecting to meet little resistance. Instead they found thousands of North Vietnamese soldiers, well entrenched and ready. The battalion was decimated. That same day, another American battalion ran into similar trouble on nearby Hill 882. General Peers and the other generals soon recognized the gravity of what we’d been trying to tell them: that the North Vietnamese had moved into the area in force and were itching to fight.

The weekslong series of engagements that followed, collectively known as the Battle of Dak To, was one of the biggest in the war, and one of the conflict’s few pitched battles. The North Vietnamese had established defensive positions on several hills, forcing the American and South Vietnamese forces to fight uphill, culminating in a horrifically bloody engagement at Hill 875, from Nov. 19 to Nov. 23. By the end of Dak To, nine American battalions from the 4th Infantry Division and the 173rd Airborne Brigade — some 16,000 men — had been committed. American bombers flew more than 2,000 sorties. The Americans eventually won, but at great cost to both sides: More than 2,100 North Vietnamese were killed, as were 376 Americans and 61 South Vietnamese soldiers.

I left the highlands in December when the offensive was all but over. I moved south to work with another team near Bien Hoa, just north of Saigon. When I got there, I saw all the same sorts of communications indicators we were picking up in the highlands before Dak To. We weren’t alone; American signals-intelligence units in the northernmost part of South Vietnam were intercepting the same patterns. We realized that an offensive was going to occur throughout the country starting at the end of January.

The N.S.A. pulled together all the evidence, and once again we presented it to the military leadership. And once again, the generals refused to believe us. At the time, Marines in the north were under siege at Khe Sanh, and the top brass in Saigon was convinced that any other North Vietnamese activity was only a diversion, an effort to pull away American forces from what they and Washington believed was Hanoi’s plan to reprise the victory over the French at the siege of Dien Bien Phu, 13 years earlier. Despite mounting evidence to the contrary, they did not prepare for an all-out assault, and at the end of January they were thrown back on their heels by the Tet offensive.

Put differently, it’s not quite correct to say that Tet caught the Americans and South Vietnamese by surprise. The intelligence was there, and the recent experience leading up to Dak To should have persuaded General William Westmoreland to take it seriously. Instead, he chose not to believe it.

The problem was bigger than Generals Peers and Westmoreland; eight years later, the same mistake was made with the fall of Saigon. By then I was the N.S.A. station chief in the city. I warned Graham Martin, the American ambassador, about overwhelming evidence showing that Saigon was about to be attacked. He refused to believe me and didn’t call for an evacuation of the thousands of American civilians still in the city, along with our South Vietnamese counterparts. When the North Vietnamese attacked a few days later, the city descended into panic. I escaped under fire. My South Vietnamese partners, the men I was working with, weren’t so lucky. Some 2,700 of them were killed or captured and sent to “re-education” camps.

Was Cassandra blessed or cursed? Those of us who worked in intelligence during Vietnam know the answer.