I have a game for you. Get one hundred things--pennies, toothpicks, whatever--and set them up in a line, from left to right. This is our voting populace in simplified, one-dimensional form. Each "voter" has just one simple rule to decide how to vote: vote for whichever one candidate stands closest to them (yes, we're using plurality voting; and if there's a tie for closest, each gets a half vote). Now, grab another thing--a pencil, a button, it doesn't matter--and place it on the line, between two voters, so that 17 voters are to the left of it. This new thing is one of our "candidates", Libram McLeftyson. Nearly five out of six voters think he's too liberal. Grab another candidate, and place it, mirror-like, so that 17 voters are to the right of it, and in equal measure to her competitor, nearly five out of six voters think Constance O'Righterly is too conservative.
Now, here's how you win the game: place a third candidate between any two voters (except right on top of one of the existing candidates) such that they win the election. Go on, try it; I'll wait while you count your things.
Give up yet? You should, because the challenge is impossible. No matter where you put your third candidate, it is impossible to make it so they win; even putting them right in the center won't do it. As an added bonus, you'll find that whichever of the other two candidates you put them closer to, also loses. And if I move them from inside the 17 marks to precisely the 25 marks, you can't even come in second. I can even put them at the 49 marks, practically next to each other, and there's still nowhere you can go where you can win.
More and more Americans are agitating for a "third party", most without even realizing that there are already dozens of active third parties to choose from. But they don't win. They can't win. And this is without even acknowledging the effects of tactical voting, this is with honest voters! In this simulation, the only ones voting for the big-two are the ones who honestly believe them to be the uniquely best option available.
The existence of a multitude of third parties hasn't change this. Screaming that we need even more third parties won't change this. You cannot win this game. The only way you can win, is if you change the rules.
And the rules that would give third parties a chance to win are approval voting and score voting.