Our democratic system was originally setup to involve as many parties as possible. In fact, George Washington warned in his farewell address about the establishment of parties at all and advised that they not be formed. In practice, however, two parties were able to dominate by the mid-19th century.
Today, the most viable option for third parties exist in states that allow candidates to run on multiple party tickets. A great example of this is the Working Families party in New York State. The Working Families party nominates candidates, usually on the democratic line, who can be voted for under the Working Families ticket instead of Democratic -- to give the politicians the message that they care about social issues for working families. The party is able to exert significant influence by refusing to endorse candidates who do not stand up for working families -- and has cost several candidates their seats because of their failure to earn the party's endorsement.
Note that these third parties have primaries of their own that are sanctioned by the state.
That said, parties like this are the ones paving the way towards BALLOT REFORM, which if instituted would open up the field to more parties who can gain real influence. Vermont or New Hampshire are seen as the states most likely to institute this first.