Excluding the Darién Gap (which you'd have to take a ferry across), is it possible to travel from, say, New York, NY to Buenos Aires, Argentina via train?
No. Wikipedia has a page on rail transport in Central America with an overview of the situation in each country. The Man in Seat 61 also has a page about Central and South America (though it does look somewhat incomplete, at least for South America).
Overall, each country has, or doesn't have, its own network. There is currently no railway crossing any border anywhere between Mexico and Columbia, at least. According to this map, in South America, the only international connections are between Bolivia, Argentina and Brazil (neither of the two links between Bolivia and the Chilean coast, to Arica or Antofagasta, carry passengers, and the Arica line is currently closed).
If you want to go from Mexico to Panama, railways won't help you much. Guatemala has none. El Salvador has only a few commuter trains that would save you at most a few miles. Honduras has a few goods trains and commuter trains but there too I don't think any are helpful to cross the country. Nicaragua is another trainless gap. Costa Rica currently only has a commuter line around San José. In Panama the only railway is the one that runs parallel to the Canal.
From Colombia to the South, you'll need to go via Peru or Venezuela. In Colombia, the existing tracks wouldn't help you much, and anyway the only passenger train is in the Bogotá region. Venezuela has a few trains but there are still large gaps if you want to reach Brazil, and then the only trains in the north of Brazil are on the coast. In Peru, finally, there are trains that will take you part of the way between Ecuador and Bolivia, but with several large gaps. You can go from [Cusco to Puno](http://www.seat61.com/Peru.htm#Cusco to Puno) on the shore of Lake Titicaca, then a ferry to Guaqui in Bolivia (if it's currently running — I can't find reliable information) and there a train to La Paz. Bolivia has several passenger networks, but here too there are gaps between these networks, and there are no tracks connecting the eastern and western network. The western network runs to Villazón on the Argentinean border. The Argentinean network no longer runs to La Quiaca, so you'll need a combination of buses and optionally a tourist line, the Tren a las Nubes, in Salta, to reach the main network at Tucuman.
To the best of my knowledge it's not.
The passenger train service in Mexico doesn't seem to have long distance trains and in the United States you can get to the Mexican border in El Paso or San Diego but you never actually cross it in a train. So your trip is going to be cut short long before you get to the Darien Gap.
Author Paul Theroux tried it in the 1970's, as documented in his book 'The Old Patagonian Express' [ISBN 0141189150] - I think he had quite a few gaps though, and used several lines that are no longer in use...
On that basis, I'd say that no, it isn't, and it is in fact less possible now than it was 30 years ago.