Does "incumbency advantage" actually exist once a party's been in office for more than one term?
Good question. Let's pull out the relevant cases. I've set a limit at the beginning of the fourth party system (1896), when campaign methods underwent significant modernization. Green items mean the "incumbent" party held the White House, while orange items indicate the incumbent candidate lost.
1904: T. Roosevelt running for first full term after McKinley's assassination
1908: Taft running after T. Roosevelt's extraordinary popularity191219201928: Hoover running at a moment of unprecedented prosperity1932
1940: FDR
1944: FDR
1948: Truman with the upset1952
1960
1968
19761988: Bush benefited from Reagan's massive popularity1992
2000
2008According to this rough picture, it seems to me like the party "incumbents" seeking a third term for their party are actually at a disadvantage, unless the precedent they are exceeded was extraordinarily popular.