At one point Robert McNamara tells President Johnson that there are 13 US battalions in Vietnam, and goes on to say this is 51,000 troops. This would mean approximately 4,000 troops per battalion. Given that a US battalion would only have 500-800 troops he is actually talking about 13 brigades (each containing several battalions) and not 13 battalions.
The movie implies LBJ presented the Medal of Freedom to McNamara prior to the Tet offensive. In fact, Tet began on January 31, 1968 and the ceremony of the award presentation took place on February 28 of that year.
Lyndon B. Johnson was a native Texan. However, Michael Gambon's native British accent occasionally slips in, particularly in the pronunciation of some words ("taught", "fought", "should" or "heart") and the use of some terms that a would be unfamiliar to an American, such as "goobledygook".
The teleprompter visible every time Johnson gives a speech in congress is way too modern. Not the one ones from the 60's.