For one thing, DNA and other evidence strongly indicate that the present day Native Americans started out somewhere in Siberia well before 20,000 years ago. Both the genetic and linguistic diversity of the Americas and the fact that Native Americas are recognizably different in appearance from any modern Asians support an early separation. There just isn't any way they could have shown up in the Americas as recently as 9000 years ago.
For another, although there is also genetic and linguistic evidence for direct connections between peoples further south along the Asian coast, like the Japanese, and certain peoples on the Pacific coast of the Americas, that evidence suggests a trans-Pacific migration well after the original settlement from Siberia. For example, this is the opinion of linguist Johanna Nichols:
"There was probably a second influx, she added. There is a narrow strip of different language families along the western coasts of the Americas that matches patterns found only in other Pacific Rim nations.
" 'They are 12,000 years old, but certainly not 40,000,' she said."
http://www.trussel.com/prehist/news52.htmhttp://scicom.ucsc.edu/SciNotes/9901/echoes/echoes.htmIn short, I'm prepared to believe that these long-headed groups arrived in the Americas by sea towards the end of the Ice Age -- but not that they were the original settlers.