The last time the US saw a truly multi-party election was 1860. There were six parties running presidential candidates and the electoral vote was split between four of them. That election was an immediate precursor to civil war. Since then various arrangements and factors have worked to create am enduring two party duopoly. One result and arguable benefit of this has been a high level of political stability. Many other countries have multi-party systems. They seem to contribute to an increased level of political instability in most cases. Depending on one's point of view that may or may not be a bad thing, but it is a difference from the US.
There are structural political arrangements that work to exclude third parties from attaining political leverage. The principal one is the winner take all laws that govern the allocation of state electoral votes. Maine and Nebraska are the only exceptions to this. While this is a matter of state law rather than federal constitutional law, the courts have never questioned their constitutionality. It is also very rare that anybody ever gets elected to either house of congress who is not a candidate of one of the two major parties.
There have been a few bumps in this generally level road. In 1948 and 1968 southern segregationists mounted third party challenges and because of their solid regional base managed to capture a bloc of electoral votes. Ultimately their cause did not prevail.
The 1992 presidential election saw Ross Perot mount an independent challenge to the major party candidates. This came in the midst of a high level of populist angst over a major recession. Perot ran a bizarre on again off again campaign and was still able to get almost 20M popular votes. He got no electoral votes. To speculated about the impact he had on the outcome requires an assumption as to whether he took more votes away from Bush or Clinton. Regardless it is difficult to argue that Perot had any lasting impact on programs and policy.
In countries with multi-party parliamentary systems small parties can hope to join a coalition government and receive some ministerial post in the bargain. The US system doesn't offer such possibilities. In the highly unlikely event of a third party candidate being elected to the presidency, he or she would have absolutely no base in congress. When a president of a major party loses the congressional majority that party still has a substantial minority which leaves room for horse trading on specific issues.
The two greatest political upheavals in American history have been the civil war and the great depression. The conflicts that led to the civil war did create a fundamental restructuring of the party system and brought the fledgling Republican Party to power. However, the great depression only ended the position of the Democratic Party being out of power. To gain the power that pushed the new deal through required a coalition of northern labor activists and social democrats with conservative die hard segregationists. But it all happened within the two party system.
I am among those people who would like to see some fundamental change in prevailing US economic policy. However, I don't really think that it is going to come about anytime soon. If it does happen it would have to be as a result of a policy shift within the Democratic Party. That has happened before and it could conceivably happen again. However, I do not think it will come about at the hand of a third party. History and the prevailing political culture are fundamentally stacked against it.