In conditions of high infant mortality, the great majority of people do not leave any descendants beyond the second or third generation. So if genes can be traced from 80 people among the original "Native American" colonists, then the total number who crossed the Bering Straits would have been very much larger than this.
Having said which, people can get away with a lot of inbreeding before it gets really damaging. Consider for example the population of Pitcairn Island, descended from 27 people (9 European and 18 Tahitian) who arrived in 1790. There were 15 men total (9 European and 6 Tahitian), but today only 12 surnames, only one of which is Tahitian. There are 16 Warrens, descended from a later arrival in 1830, and 15 Christians, descended from Fletcher Christian the leader of the mutineers.
Of course the Pitcairn settlers had plenty of genetic diversity to begin with, being of two very different races.