Have the landmasses of North and South America ever been part of the same continent?
Apart from today, yes.

Back in the Cambrian (~500 million years ago), there was a continent in the southern hemisphere called Gondwana. It was a combination of several modern day continents: South America, Australia, Antarctica and Africa. A small piece of North America that corresponds mostly to modern day northern Canada existed in the northern hemisphere, called Laurentia. Curiously, some parts of northwestern Europe were part of that as well (Scotland, Norway).

Between the Devonian (~370 million years ago) and the Triassic (~200 million years ago) almost every continent on the Earth was amalgamated into one super continent called Pangaea, including Gondwana and Laurentia. During the Jurassic and Cretaceous (~150 million years ago) the entire thing separated due to the opening of the Atlantic Ocean, sending North America north west and South America south west. Since then the two continents were separated, until very recently (geologically speaking) that a land bridge formed between the two.

A very good website with reconstructions of paleogeographical maps is this one:

http://scotese.com/

Note that these are all interpretations and they are open for debate. But the general idea is correct.