Today Senator Elizabeth Warren released results of a DNA test showing “strong evidence” of Amerindian ancestry. This ancestry has been reported as fact by several media outlets. But in short, this report does not prove Amerindian ancestry, owing largely to the very small amount Warren's test suggested.
The controversy over Elizabeth Warren's ancestry stems to back to 1984, where she supplied recipes to a cookbook under the title, "Elizabeth Warren, Cherokee". She would later appear as a minority law professor in an Association of American Law School's directory; and later still be identified as a Native American at the University of Pennsylvania. Though at other times during this period Warren identified herself and was identified by others as white. Later still she was identified as a Native American by Harvard up to 2004.
Warren's assertion that she is a Native American stemmed from family lore that said her mother's family had Cherokee and Delaware blood.
This claim and history has been seized upon by critics, most notably President Donald Trump, who uses this history to mock her. At a recent rally he quipped,
I have more Indian blood in me than Pocahontas [Elizabeth Warren]. And I have none. Sadly, I have none.
Trump went so far as offering to donate $1 million dollars to charity if Warren took a DNA test and it proved she had Indian ancestry.
The results of Warren's DNA test assert a portion of her DNA that is 13.4 centiMorgans of her tested DNA “strongly suggest” Amerindian ancestry. In layman's terms this means Warren has somewhere in the region of 0.38% Amerindian ancestry, which can be interpreted as a great-great-great-great-great-great-grandparent. In Warren's case this would point to an ancestor who was probably born in the first half of the 18th century. However, due to the random inheritance of DNA it could be up to two generations earlier or later.
The problem is that detecting ethnicity from DNA is currently rudimentary and can produce sometimes highly inaccurate results. Testing your DNA with various available methods can lead to widely different ethnicity results. By and largely such tests are only generally accurate at a continental level and to a lesser extent regional level, e.g. West Asia. Determining at a national level is currently problematic. Results from any ethncity DNA test should be taken with a shaker of salt.
Further incidences of low amounts of a certain ethnicity are typically ignored by professionals due to the temperamental nature of the tests. Of the now cira ten million people who have taken genealogy DNA tests, the subjects regularly find sometimes several or more percentage points of ethnicities they were not expecting. This includes results from continents they have no ancestry from. Such incidences often occur under 1%, as in Warren's case. Professionals view such incidence as possible, but more likely to be a false positive produced by a developing science.
The questionable nature of Warren's results is further compounded by the very small sample of ethnically assigned DNA it was compared to. It was compared to just 148 DNA samples. The largest DNA tester, Ancestry, currently uses over 16,000 samples.
And despite that large pool Ancestry's results are still temperamental, particularly outside of Europe and is susceptible to providing false positives under or around 1%.
If you take multiple DNA tests, you will be given a number of false positives. I have tested my DNA with various services and tools and have received a number false positives, including North Africa (over 2%), Melanesian (<1%), Tukric, Italian (2%), Swedish (5%); but most notably a number of tests, particularly MDLP return me as high as 3% Amerindian. Searching forums and blogs you will find plenty of people scratching their heads over completely unexpected results such as these.
To further show the temperamental nature of these tests, Ancestry, the company that has provided more tests than all other combined has has a page in it's support section explaining why siblings results might be so widely different. It provides an image showing their ethnicity projections for four siblings:

As you can see there is some considerable difference and notably one has 4% Eastern Europe, while the others have none. With the level of data for those siblings, the 4% would be considered a false positive.
It is possible that Elizabeth Warren's <1% of Amerindian ancestry suggested by a low sample DNA test could be evidence of Amerindian ancestry, but the typical interpretation would be to consider it a false positive.