I'd imagine this is possible, given a birth date circa 1900, possibly with lying about age involved. The DoD considers November 1, 1955 the official start date of the Vietnam Conflict, since this is the date MAAG Vietnam came into being.
Gen. Samuel Tankersley Williams
Fought in France in WWI. Participated in Normandy in WWII. Commanded the 25th infantry division in Korea. Commanded the Military Assistance and Advisory Group in Vietnam from 1955 - 1960.
Edit: found another candidate.
France in WW1, General for several campaigns in both Pacific and Europe in WWII, Army Chief of Staff for the Korean War, Special NATO adviser to Vietnam 1953-1955. The last role was more of an ambassador position than a soldier position and might technically have ended before the war began.
Col David Hackworth fought in the latter three, so that may be a start for further research.