I know of ONE time in history when a third party became a major party. In the 1850s the Whig Party disintegrated, and the Free Soil Party absorbed enough Whigs and anti-slavery Democrats to take the place of the Whigs. At that point, its members felt it needed a new name, and because nobody else was calling themselves “Republican” at that point, they went with that. Therefore, in order for the Tea Party to succeed, either the Democrats or country-club Republicans have to go.
The freesoil party is a good analogy. They elected a fair number of congressmen but were most effective in attracting disaffected Whigs and Democrats into a new party (Republicans).