Yes. In 1872, the Democrats did not run a candidate. Instead, the major candidate against Ulysses S. Grant's re-election campaign was Horace Greeley, a member of Grant's own Republican Party, who joined a group of Liberal Republicans to run a more liberal alternative to Grant. The Democrats endorsed the long-shot Greeley instead of running their own candidate, who was certain to go down in defeat.

A similar thing happened in 1912. Taft ran for re-election as the Republican candidate, but the previous Republican president, Teddy Roosevelt, challenged him at the convention, and failed. Instead of giving up, Roosevelt formed his own third party and ran against Taft once again in the general election. Both Taft and Roosevelt lost to Woodrow Wilson.

Also, notably, a few vulnerable candidates faced successful challenges in the nomination process, rather than the general election process. These include John Tyler, Millard Fillmore, and Lyndon B. Johnson. All three had succeeded to the presidency upon the death of the previous president, though LBJ then won his own term outright. Tyler was kicked out of his party, so he formed a third party, but dropped out in August 1844 before the general election. Fillmore went to the Whig convention in 1852 trying to get the nomination but lost it to Winfield Scott. LBJ declined to run again in 1968 when it became obvious he would struggle to get renominated by his party, having won the New Hampshire primary by a narrow margin.