Exceptions occur if there is no "obvious" establishment candidate or "heir apparent" -- 1992 was a free-for-all, for example, which Clinton handily won as a skilled campaigner -- or if they're seen as inherently weak. Obama won over the party apparatchiks in 2008, for example, who switched from Clinton.
But sitting presidents are rarely primaried even when they're vulnerable, and the party can effectively shut down that process to protect their candidacy. Everybody is scared of 1968.