I can’t think of any particular significance to the figure aside from its sheer size, but 5 August 2007 is the 1,600th day since the invasion of Iraq to stop Saddam Hussein from turning over WMDs that didn’t exist to al-Qaeda terrorists he had no operational contact with.
The address of the White House is “1600 Pennsylvania Avenue”.
1,600 kHz is the top end of the commercial AM radio bandwidth.
In the year 1600, William Shakespeare’s A Midsummer Night’s Dream was first performed and the future King Charles I of England was born.