This answer does not talk about history; it is only about what stops this from being a possibility now and ever since we got to the point where the only parties with any power were Democrats and Republicans.
It takes way too much support in the system of how the U.S. is set up for this to ever be possible in the foreseeable (or really any) future.
The U.S.'s electoral systems are really just a large number of very separate elections.
Legislative, presidential, etc.
elections all divide votes into groups with no bearing on each other: representatives in districts, senators in states, and the president also in states due to the electoral college.
This means that, for example, if in an election for a state house district the Democrats get 50.001 percent and the Republicans get 49.999, the fact that the Democrats only got .001 points more than the Republicans does not mean anything; having even a margin of victory of only 1 vote does not get reflected in the current system. The seat is given to the Democrats, and the votes for the other party doesn't do anything.
As the two major parties are so large and this system means that even almost getting the most votes does not award you anything, structurally it is essentially impossible to become bigger than both of them in any meaningful election at this point. The systems are all winner-take-all per district/state*, making it so that even a party that gets a great third or second place in a large number of elections will have nothing to show for it.
Because the two major parties are so giant and the system is winner-take-all for all elections, it is impossible for another party to get enough votes to gain any representation.
*except for Nebraska and Maine in presidential elections; I do not know of any state legislatures with more proportional systems.