NO. Ramming is considered to be extreme unsportsmanlike behaviour in NASCAR, and what actually goes on in the track has nothing to do with what happens in Days of Thunder. What DOES happen is bump-drafting, which is something totally different. In bump-drafting, the car behind "pushes" the car in front with its front bumper, so that both cars can travel faster along a straight. Doing the same thing on a turn is dangerous, so the car behind will usually back off just enough to avoid contact, and wait until both are on a straight again.
Another technique utilised by NASCAR drivers is the "bump and run", which consists in the car behind tapping the rear end of the car in front so it gets lifted off momentarily, normally during a turn. The driver of the car in front will need to make corrections and will thus lose speed, making it so that overtaking is much easier for the trailing car. It is seen as dirty, unethical driving, though. With the new COT cars, this maneuvre has become a lot more difficult to do, as the COTs have a lower rear end and a higher front bumper.
However, outright ramming like in Days of Thunder is not acceptable, and even punishable.