I did a bit of research to see what history might teach us, and it wasn't a whole lot of help to answer the question posed in this thread.
HISTORY OF POLITICAL PARTIES
In the 1790s, the Republican Party was led by Thomas Jefferson and James Madison; while Alexander Hamilton led the Federalist Party. The Federalists were for a centralized national government while the Republicans were for limited government, especially in respect to commerce.
In 1796, John Adams, a Federalist, was elected President and Thomas Jefferson, a Republican, was elected Vice President. As that didnt work out well, we soon had a 12th amendment requiring both the President and Vice President to be from the same party.
The last Federalist was James Monroe elected 1816. By 1828 the Republican Party had divided into the Democratic Republican party led by Andrew Jackson and the Democratic Republican Party that eventually became the Democratic Party making it the oldest political party still standing, and John Quincy Adams headed the new National Republican Party that represented the old Federalist Party.
In 1832, a group of Democratic Republicans, resenting Andrew Jacksons affiliation with the Masons, created the first third party and and called it the Anti-Mason party. At the same time the National Republican Party was dissolving as Henry Clay and Daniel Webster led development of the Whig Party. In the 1840s, the Whigs elected William Henry Harrison and Zachary Taylor.
About the same time, the Free Soil Party was formed by supporters of the Wilmot Proviso, which prohibited slavery in any territory acquired from Mexico. In time, another party, the Know Nothings, developed from Democrats who did not like the wave of Catholics immigrants.
On July 6, 1854,. The Republican Party was formed in 1854 by the Whigs and Free Soil party. Eventually, the Know Nothings dissolved with most members joining the Republicans. At their first convention on July 6, 1854, the platform was one that called on the country to abolish slavery. That emphasis continued until accomplished by the Civil War under the Presidency of Abraham Lincoln.
Thus, as political parties form and dissolve, we are now left with two viable political parties called Democratic and Republican. Parties. Over time many more parties have been formed, each one to further their own agenda or leadership that they felt not adequately represented by the two major parties. None have ever gained enough traction to be viable.
And the question remains now whether a third party could be viable and whether it would a) most hurt the Republicans or b) most hurt the Democrats more.
(Most adapted from article in The Examiner.
History of political parties in the USA)