TBogg: Sure, he wasn't having multiple affairs while consorting with his mob friends and living large, but George Bush offered him a job anyway:

Democratic Sen. Joe Lieberman has twice in recent days said "no" when approached about the possibility of a major job in the second Bush administration, CNN has learned.
The Cabinet vacancy at the Department of Homeland Security was the subject of the latest overture, according to congressional and other government sources. Those sources said the earlier overture was to see whether Lieberman might be interested in becoming the U.S. ambassador to the United Nations.

Thus President Bush was foiled in his attempt to further diversify his Cabinet by adding an invertebrate.