Recent DNA testing shows that "...about a third of the ancestry of today's Native Americans can be traced to 'western Eurasia,' with the other two-thirds coming from eastern Asia..." Source: Science Magazine.
"[This discovery] also implies that traces of European ancestry previously detected in modern Native Americans do not come solely from mixing with European colonists, as most scientists had assumed, but have much deeper roots."
Comment: Genetic tests have apparently determined that Native Americans are one-third European [Ibid]. According to a 2012 article in Science Daily, "[t]here is a genetic link between the paleolithic population of Europe and modern Native Americans. The evidence is that the population that crossed the Bering Strait from Siberia into the Americas more than 15,000 years ago was likely related to the ancient population of Europe."
These genetic similarities appear to support the Solutrean Hypothesis -- the theory that the first Americans were Stone Age Europeans who crossed the Atlantic to North America thousands of years before the Siberian ancestors of modern-day Native Americans, only to disappear through war and/or interbreeding. The reality may have been more complicated than that, however.
The current discovery described in the Science article tells a different story and goes a lot further back in time. It suggests that some of the ancestors of the proto-indians who crossed the Bering Land Bridge into North America may have been themselves descended from earlier groups living in Eurasia, which includes the area we now call Europe.
The still-hazy picture which is starting to emerge from all of this shows waves of hunter-gatherers migrating east from Europe and Asia and eventually crossing over to North America, following the game herds across a landscape of advancing and retreating glaciers. Northern Europeans could have made the sea (or ice-bridge) crossing west to North America as well, so the Solutrean Hypothesis could be accurate, but incomplete.
One way or another, the traditional story about how the continent was originally settled is rapidly becoming obsolete. North America was populated by successive waves of migration involving different groups of people. At the moment, there doesn't seem to be any way to know who the First Americans were or where they came from, but I expect that North America has been inhabited for a lot longer than currently believed.