No, though many presidents were previously governors (including Ronald Reagan, Bill Clinton, and George W. Bush). Many other presidents were senators or vice-presidents.
Rockefelller and Clinton
During Thomas Jefferson's first term, the Vice President was Aaron Burr. Jefferson's second term as President was George Clinton's first term as Vice President (Clinton was one of the two U.S. Vice Presidents who served under two Presidents).
Former President Bill Clinton doubtlessly had more in common with his own vice President Al Gore, than he did with former Presidents Nixon or Ford.
Both of James Madison's Vice Presidents, George Clinton and Elbridge Gerry, died in office.
Clinton had one VP for his entire presidency: Al Gore.
The president was Bill Clinton and Al Gore the vice-president in 2000.
Former Vice President George Clinton is not related to former President William (Bill) Clinton.
Yes
George Clinton and Elbridge Jerry (both died in office)
James Madison lost both of his vice-presidents to death. They were George Clinton and Elbridge Gerry.
Al Gore served as the vice president under President Bill Clinton from 1993 to 2001. Therefore, the president at the time of Vice President Al Gore was President Bill Clinton.