There’s a bit of rhetoric around the dominance of the two major parties in the United States, that treats it as some sort of cultural issue where if only people would break out of their brainwashing and VOTE for Third Party candidates that dominance would be shattered.  This is unfortunately untrue.  Breaking the Sith-like “Rule of Two” that dominates American politics would take bedrock Constitutional reform to the election process, that is grossly unlikely because it would require the states to willingly sacrifice their power in said reforms’ ratification.
Part of the misnomer is the idea that the system looks the way it does because the two parties rigged it in their favor.  Certainly there are aspects here and there that do reflect that two parties making it harder for Third Party or Independent Candidates, but these issues are minor in comparison to the forces that created the dominance of the two major parties in the first place.  Ironically, it is the fact that our Constitution was NOT written with national parties in mind that creates the issue.  Those representative and parliamentary democracies that support a wide range of parties have electoral systems that are built around the assumption the political parties, while the US system was built around the assumption that there wouldn’t be any.
The Framers believed that state loyalty would trump all in national elections, and that political interests between the states were too diverse for parties to organize around national agendas.  Obviously they  would prove themselves wrong as they gathered into camps of Federalists and Jeffersonian Republicans ridiculously quickly, because of the electoral forces they built into the system.
Election laws for federal office are (for the most part) set by the states.  The states determine the process of electing Congressional Representatives, Senators (originally appointed by state government, but after the 17th Amendment, required to be by popular vote of state residents), and how to distribute their electors for the electoral college for President.  Overwhelmingly, in the interest of simplicity, and (in the case of the electors) to maximize the political influence of the state, they favor a system of plurality rule/winner-take-all.  In UK parlance, this system is also called “First Past the Post”.  It simply means that which ever candidate gets the most votes, 100% wins.
Sounds simple enough, right?  The candidate who gets the most votes wins.  What’s wrong with that?  Seems a natural and self-explanatory enough system for a country that uses “Single Member District” representation (as opposed to systems of proportional representation, like many multi-party parliamentary countries - I’ll describe that later if you’re curious).  Well, it is - that’s why it’s the default.  The “problem” arises in that it doesn’t require a majority - just the largest plurality.  If a majority was required to win, we’d have a lot more run-off elections for Congress, as the largest vote-getters fought it out in revotes trying to get a true majority (50% +1 vote).  Notably Louisiana uses a majority rule system for its elections and has a TON of run-offs compared to most states.
So, why does this result in 2-party rule?  The reason is simple.  Lets look at Presidential elections.  California has 55 electoral votes, the most in the country.  Under our system of two party rule, it’s a solid “Blue State” meaning it is reliably going to vote for the Democrat in the Presidential race, and as such it doesn’t attract a lot of attention in the race, because the candidates are going to focus their resources on states they can win or are at risk of losing (the so called “Swing States”).  Let’s assume that wasn’t true though.  Lets say there are 5 political parties reflecting the variety of views and agendas in play in California and no two hold solid dominance like in our system?  What happens under “first-past-the-post”?
Hypothetically…
A right-wing party gets 25% of the vote
A socially-liberal free-market libertarian party gets 17% of the vote
A moderate progressive-capitalist party (ala the Democrats) gets 24% of the vote
A socialist party gets 17% of the vote
A break-away environmentalist party (that feels the socialist party above puts too much emphasis on jobs at the expense of conservation) gets another 17%
So what happens? Â The right-wing party, with 25% of the vote gets all 55 electors, despite the majority of voters being vehemently opposed to their agenda. Â This creates the inevitable incentive for opposition to that ruling plurality to unify together, despite their differences, for them to be the ruling plurality. Â That inevitably reduces the number of parties down to the rule of two. Â Any remotely successful third party inevitably aides the largest party they are ideologically FURTHEST from, under this system.
Even the so-called “proportional” distribution of electors that a few states use doesn’t fix this problem.  Because electors are elected by individual districts in the state using the same system only more localized.  This opens up a bit more possibility for a few third party electors to actually go to the electoral college, but doesn’t resolve the overall issue.  (This is more or less why there are comparatively more successful third parties in the UK.  Because they don’t vote nationally for Prime Minister, instead each district elects their MP using the first-past-the-post system, and whichever party has a majority of seats in Parliament, their party leader becomes Prime Minister.  Thus you have openings for more localized two party contests that aren’t the larger dominant two-parties.  The system still overwhelmingly disadvantages third parties, as any Lib-Dem will talk your ear off over).
The states (who control the ratification process for Constitutional Amendments) are in no rush to change this system, as it assures their importance in the Presidential election.  Lets say instead we had national popular vote determine the President, requiring majority win, with a run-off between the two top vote-getters in December if no one wins a majority in November.  Now third parties can be more competitive, assured that even if they can’t win the Presidency, they aren’t helping the worst possible option, and can always vote for the lesser evil in December, thus letting them build the groundwork necessary to be competitive in Congressional races.  What happens to states like Ohio in this scenario.  Currently they attract overwhelming attention as the Presidential candidates battle it out to eek out a plurality win and thus snatch its 18 electoral votes.  But under this alternate system, the difference between narrowly winning vs narrowly losing Ohio barely matters, and instead Presidential candidates are going to focus on highly dense population areas where they can get out the largest vote of their supporters.  Maybe good for California, New York City (though upstate will be ignored), and parts of the South, but overall you can see why state governments would not be happy at all with this scenario, as Senators hoping to run for President one day stop trying to shower them with goodies like farm-subsidies to make their voters happy.
So in short, no, you aren’t sticking it to two-party rule by rebelliously voting third party.  You’re just making one of the two snicker.  Opening up our elections isn’t a matter of cultural change, it’s a matter of legal reform to the very foundation of how our elections work.  And that reform is next to impossibleÂ