The Vietnam war is one of those wars that nobody really remembers the point of. Primarily I suspect in America because the majority of the focus is on the anti-war movement and counter-culture. Vietnam conjures up imagery of David and Goliath, a technological superpower being defeated by a gun made of wood that worked in the mud. However in essence the Vietnam war was a war of attrition.
So first and foremost we need to understand the players, you had the tangentially communist bloc supported North Vietnamese and the directly US supported South Vietnamese. Now I can't stress enough how important the South Vietnamese are to the ultimate failure of the Vietnam war. But we'll get back to them later. Lets talk attrition.
Major US involvement started in 1964 however US forces had been involved in what was then French Indochina since 1950 Vietnam at the time was then French colony and France itself was fighting its own "Vietnam" to keep its colony which was being bankrolled by the US as part of America's policy of containment.^(1)
The forces battling the French were "led" by Ho Chi Minh (the reality is complex but for the purposes of this post we're going to focus on Minh) the Viet Minh as they were known at the time were a communist group that in 1945 aided the French in suppressing nationalist movements before turning on the French and starting their own war of independence. The importance here isn't on who was battling who but the start date. 1945^(2). To Minh and the communist North this was the start of their war. From the onset Minh's strategy was to bleed his enemies dry. According to HR McMaster^(3) Minh stated a french diplomat " You can kill ten of my men for every one I kill of yours. But even at those odds, you will lose and I will win " Again this was a quarter century before the US' direct involvement and Minh and the North Vietnamese were taking a meat grinder approach to independence.
So back to 1966 "what was the US war plan?" well you have to remember the US did not want to (and does not like to be) seen as an "aggressor". The US could not just go "conquer" a country to prevent it from spreading communism during the cold war. This is where South Vietnam comes in, the reason for the US involvement in the country was to "prove" it could prevent another country from turning to the communist bloc^(4). And it found the pretense it needed to do so by "aiding" the South Vietnamese regime. Kennedy at the time had lost several major "battles" to the communist bloc and South Vietnam was to be his "line in the sand"^(5) A sentiment that Johnson continued. However, If the US were to "invade" North Vietnam they would no longer be merely protecting South Vietnam but seen as an invader which could set off the USSR and China and defeat their entire purpose for being there. It's also important to remember that initially the war was primarily focused on the US side as being against the Vietcong, an insurgent group technically separate from North Vietnam with its own military units and which primarily recruited in South Vietnam.^(7) This meant that at the onset the US was not actually fighting "against North Vietnam" but against the Vietcong as well, invading the North wouldn't defeat the Vietcong and it wouldn't achieve the US stated goals of propping up the South Vietnamese government . The war plan that the US eventually came up with was to basically call the Vietcong and North Vietnamese bluff and to try and see how long they could keep up the meat grinder. ^(6) Us command basically figured that eventually if they just kept killing Vietcong that there would be no more insurgents left. This was evidence by the highly publicized "body count"^(8)
So yes to put it mildly the strategy was indeed to just "let them keep attacking until they gave up" which again seeing as how the North Vietnamese had been fighting this "war" for nearly 3 decades, wasn't going to work.
There are other reasons the US "lost" but that should suffice it to answer your question.
1- BBC News. (2019). Could Vietnam have been nuked in 1954?. [online] Available at: https://www.bbc.com/news/magazine-27243803 [Accessed 4 Mar. 2019].
2- Rosie, George. 1970. The British In Vietnam. London: Panther.
3- McMaster, H.R. (1997) Dereliction of Duty: Lyndon Johnson, Robert McNamara, The Joint Chiefs of Staff, and the Lies that Led to Vietnam, Harper Collins.
4,5- Mann, Robert. A Grand Delusion, Basic Books, 2002.
6,8 - Bray, Gary. 2019. "Body Count In Vietnam | Historynet". Historynet. https://www.historynet.com/body-count-in-vietnam.htm.
7- Pribbenow, Merle L. 2016. Victory In Vietnam. University Press of Kansas.