You're right. They were introduced to the troops in Vietnam (which actually covered from 1965 to 1972/1973). The M-16 was only issued to small amounts of troops at first. For the most part they used M-14s at the beginning of the campaign.
The actual design was created around 1954 (around the same time when the AK-47 was born). Originally it was known as the Stoner rifle. It took a long while to iron out the kinks in the design, and get the DoD to buy-off on the weapon.
EDIT:
http://world.guns.ru/assault/as18-e.htm
My original development times were a little off. Check this website, and scroll down. You'll see the timeline for the rifle. My bad for the original wrong info.