@juan31127,
juan, "fell apart and that cigar" is not a phrase.
"... her depression as her relationship with Bill Clinton fell apart" is one section of the sentence.
It refers to Monica Lewinsky, who had a relationship with President Bill Clinton, becoming depressed when that relationship started to "fall apart," a way of saying that they were breaking up.
"...and that cigar...." is another section.
Chai came up with a plausible context, could you post more? Then we could answer more definitively.