The electoral rules in this country strongly militate against such a strategy ever being successful. The reason is that the rules award victories to “first past the post”. What that means is that a candidate doesn’t have to win “a majority” of votes in his district, he only needs to come in first among all candidates.
So for the time being, third parties are unlikely to win any contests.
In addition to those rules, which are hard enough by themselves to give a victory to any third-party candidate, the rules established by the Democratic and Republican “owners” of Congress and every state house ensure that candidates from those parties are specifically exempted from “ballot access” petitioning that is required of every other candidate. That means that third party candidates already face an uphill fight just to circulate petitions according to the rules that each state sets simply to be named on the ballot. That takes time – and a lot of money that third parties just don’t have.
If we had a parliamentary Congress such as the UK, Canada and many other British Commonwealth nations (and others, of course, such as Israel), then “proportional representation” could be possible. Third party candidates could become elected simply because of clearing whatever bar was set for election across the whole country. That would make for a lot more chances of overt coalition government (sort of what already exists in the Republican Party today, in fact.) But we’re not about to try that, either – because whether it would serve the country better or not, it’s certainly not in the interests of the two major parties to even consider it. Aside from which it would tend to make government somewhat more ‘unstable’ than it already is, I think.