We have, basically, the Democrats and the Republicans. Between the two they have, pretty much, fixed it so a third party is just not gonna fly no matter what. The problem is that both parties are not really political parties but organizations which are constantly whining about their 'membership' or lack thereof.
I can remember a time when we actually had a two party system of Democrats and Republicans who were card carrying members and subscribed to the party planks. NOBODY talks about that anymore. When was the last time anybody gave a damn about party planks (the basis upon which the party is based). It would seem that means nothing. Then we came to 2016 and these two parties suddenly had leaders that were not members. The Republicans had Trump and the Dems had Bernie. Basically, anybody can call themselves a Democrat, or a Republican, and the party of choice would then embrace the claimant. This was called something like "the party tent" which, apparently, was actually a display of panicked parties grateful that anybody would even associate with them. On the Dem side the Bernie folks are actually continuing their efforts to control the party even though they didn't vote for the party candidate nor, for the most part, actually register as members.
As far as I can tell we seem to live in a very strange world where we now have parties which represent, basically, nothing and leave that stuff to be explained by them making claims to belong when they never did and still don't. The Republicans are now being 'led' by Trump and, so far, nobody actually has a clue as to what is going to happen once he is sworn in. The Dems, on the other hand, continue to wring their hands in total grief whilst demonstrating an inability to do little bit whine and are not only leaderless but completely confused.
All this is a shame. I would suggest, for the heck of it, that both parties start to specifically explain just who they are, what they stand for (planks) and insist that any candidate claiming a specific party actually be a registered member of that party. To get this done the parties need to have a core membership that understands, exactly, what the party is for and what it is against, and insist that candidates, running under their banner, not only be registered but also agrees to support the party planks (at least most). Right now, for somebody to say they are a Democrat means, basically, that they are 'liberal' (a moving target). When somebody says they are a 'republican' means, basically, that they are 'conservative' (which is just as vague as 'liberal') In other words our current political parties are actually political phantoms without a genuine membership. This is true even for those who are registered in either party. It starts with 'wings' and moves onto pure anarchy on both sides.
If either party can't get this accomplished then they are broken, should admit same, and creep off into the night. it is not an accident that there are now more 'independents' than party members and, obviously, there are good reasons for that. Our system is well and truly broken and, I believe, from the inside and its time we take a good hard look and decide how to fix it before it all comes down around our ears.