The name Flat Broke may refer money, as the Brokes are not exceedingly rich (hence their last name). The first name "Flat" and the last name "Broke" hints that perhaps Flat Broke really was "flat broke".
Life leading up to The Sims 3
Flat lives in Riverview with his wife Flo Broke, son, Skip Broke, and daughter, Ruby Broke. His brother, Buck Broke, nephew Trigger Broke, and niece, Susie Broke, live with him as well. Even though he is an Elder in the game, he looks rather different at his death, having full grey hair and no beard.
Life leading up to The Sims 2
Although Flat is no longer living at the start of The Sims 2, the Broke family tree tells us that Flat was married to Flo Broke, and that they are the parents of Skip Broke. All three of them were deceased before the start of The Sims 2's storyline, which leaves Skip's wife, Brandi Broke, widowed and raising her sons (Dustin Broke, Beau Broke, and an unborn son, which she is expecting at the start of the game), who are Flat's grandchildren.