Constitutionally, they cannot serve in a Federal and state office at the same time. And it is impossible for a POTUS to serve as his state's Governor simultaneously.
David Pryor cannot serve as US Senator and Governor of Arkansas at the same time.
Bill Clinton cannot serve as Governor of Arkansas and President of the United States at the same time.
I noticed that Al Smith was the last Democratic President of the United States.
What happened during his administration that caused the Democrats to implode?