Has an NBA game ever ended 83-81?
It has. 83-81 has been a NBA final 41 times in 73,172 games. The first was Baltimore Bullets beat New York Knicks on January 8, 1949; the latest was Boston Celtics beat Indiana Pacers on March 6, 2013.
That makes 83-81 the 598th most common final score out of 3,189 that have ever occurred in NBA. One point away, 84-81 has happened 37 times and 83-82 has happened 24 times. At one point the score disappeared for 34 years, between 1956 and 1990.