I'm currently watching a fictionalized movie called Kon-Tiki from 2012 and my understanding is that Heyerdahl manages to get from South America to Polynesia with only a raft built with technology available 1500 years ago. But what I've yet to figure out is if the only result is that this theory is plausible, or if there in fact is some substance to the theory.
What do Historians say about this?
According to linguists and anthropologists the general consensus is that Polynesians migrated from modern day Taiwan.
Geneticists have found strands of DNA that are distinctive to Native Americans at archeological sites in Rapa Nui, a small polynesian Island. However the theory that this meant that South Americans colonized Rapa Nui before the Polynesians was thrown out when it was discovered that the DNA was not present in female corpses.
It is widely debated that the Polynesians did have some contact with Native American. The Polynesians, being the great seafarers that they are may have had contact with the Chumash Indians of Southern California. As I said this is widely debated.