Let's go back a half century
for the details on how in the world a team ended up being called the Washington Bullets, of all things.
The Baltimore Bullets came into the league (then known as the Basketball Association of America (BAA)) in 1947, and were named after a nearby gun factory.
The team disbanded in 1954, but NBA basketball returned to Baltimore in 1963, when the Chicago Zephyrs left the Windy City and moved to Baltimore. The team took on the old Bullets name. In the mid-1970s, the team moved to Washington, DC, but kept the Bullets name. During the 1973-1974 season, the team went by the name "Capitol Bullets", before changing to Washington Bullets the year after.
However, due to Washington becoming infamous for its high crime, the name Washington Bullets didn't sound right. This led to the name change that ur55318008 describes above.
The Washington Bullets had some successful years in the late '70s. In 1975, they went to the NBA Finals (before being swept by Golden State). Three years later, the Bullets won the championship, defeating Seattle in 7 games. Seattle got revenge the next year (1979), shooting down the Bullets in 5 games to win the NBA title.
Unfortunately, that would be the last hurrah for those great Bullets teams of Elvin Hayes, Wes Unseld, and others. The Bullets never won another playoff series (much less compete for the NBA title) through the name change, which took effect in 1997.
Thanks to nba.com and "The Official NBA Encyclopedia" for providing background for much of the information above.