My answer is a bit more narrow than the others. Third parties in the US are rarely successful due to the first-past-the-post voting system, otherwise known as winner-take-all voting.
Imagine that a party that mostly aligns with the democrats but has some slightly different views rises up in one area that has a history of voting about 60% democrats 40% republicans in most elections. When it comes to election time they've managed to sway about half of the democrat voters and the election results come out like this:
- 30% democrats
- 30% third party
- 40% republicans
The republicans win even though 60% of voters don't want them in government, at the next election, everyone gives up on the third party and goes back to voting democrat because they have the best chance of winning, the third party all but disappears and everything basically goes back to normal.
This is known as vote splitting and leads to factional divisions and power struggles within a party in favour of forming new parties, it also means that it normally takes a massive effort to upset the status quo of a political system that uses this voting system.
If you want to learn about alternative systems of voting I highly recommend watching CGP Grey's series "Politics in the Animal Kingdom" and researching Condorcet voting.