It is a mathematically possible number but almost certainly an overestimate. I have not seen the documentary in question, but they are likely using a figure of 35 million advertised by the General Society of Mayflower Descendants ^[1][2] which would be just over 10% of the current US population.

On the surface this boils down to a math problem, where the main question is: How many new Mayflower descendents did each generation produce since they first landed in 1620? Determining that exact number is impossible, but before even trying to estimate, it's easiest to work backwards from the number we have.

Today the mean age for a woman to give birth is about 26. ^[3] So for easy math let's say that a "generation" is 25 years. If so, then the 35 million estimate implies that every 25 years, every Mayflower descendent produced on average about 2.25 new Mayflower descendents, that is, their size more than doubled each generation. ^[4] To sanity-check that number, we can look at estimated historical "crude" birth rates. That simply refers to the number of births divided by the population in a given year. At their peak in the early 1800s, crude birth rates show us that every 25 years, about 2.6 children would be born per woman in the US population. There's some debate around when and how fast birth rates fell during the 19th century, but even using high-end estimates, that number fell below 2 children by 1875. Since around 1975, the number has been at or below 1 per woman every 25 years. ^[5]

Slight aside: It's relatively easy to Google this question and find claims that a typical family in the 1800s had seven to eight children. But at least one study that cites this data is careful to point out those numbers only consider "women reaching childbearing age." ^[6] Fertility data from before the 20th century also almost exclusively deals with white families, and it's common to find such numbers described as "per household" or "per family," implying that the arithmetic likely leaves out those who didn't live in a typical white household. And most importantly, all eight of those children would not in aggregate be born before the mother turns 25.

Pre-1800 data is much harder to come by, but we can get some hints about births in early the early Plymouth colony thanks to sporadic official records and anecdotal data. Historian John Demos compiled such data and summarizes typical family sizes by pointing to a census taken of Bristol, Rhode Island in 1689:

These figures suggest a rough average of six persons per family in the Bristol population of 1689. Closer examination shows that households of four, five, and six persons were most common, comprising some 47 per cent of the whole sample; 17 per cent were smaller than this (one to three people each), and 36 per cent were larger. These results may seem surprisingly low, in the light of traditional notions about the size of the colonial family, but Bristol was not an unusual community. As I have tried to show elsewhere in greater detail, the families of this town may have been slightly larger than the norm for some other parts of the colony. But if so, the discrepancy was extremely narrow, and for our purposes the Bristol materials should be regarded as typical. ^[7]

Here is a table with the census data, and for reference here is a table from the same book that summarizes the "size of families." That table shows an average of seven or more children, but note that the sample only includes families where both parents lived "at least to age fifty." Hence the reason the census data does not reflect the same average.

There are too many unknowns here to try to derive crude birth rates for comparison above, but at least we are in the same ballpark, and there's no indication early families would have been significantly larger than the 1800s peak. In fact Demos tells us that about 75% of children born in the Plymouth colony reached age 21 (the number today is 99%), making it less likely enough women reached childbearing age to match the 1800s peak birth rate.

Even if we assume the peak birth rate of 2.6 children per woman per generation applied all the way from to 1620 to 1800, we still confront the reality that the rate fell significantly after 1875. Using these very round numbers, a more reasonable high-end estimate would be on the order of 5 million living Mayflower descendents.

But what if we're splitting hairs? We're talking about birth rates of 2.6 that slowly fall down toward 1. The estimate of 2.25 implied above doesn't sound that far off. The problem is, we're making one other HUGE assumption. For the 35 million number to be correct, we need to apply that birth rate to every Mayflower descendent. That means we're assuming everyone had children with someone who was not also a Mayflower descendent. For example if two Mayflower descendents have two children together, in that family the next generation didn't double in size, it stayed the same. In genealogy this is called "pedigree collapse" and it is a major mitigating factor on the number of new descendents each generation. ^[2] In particular, in the early days of the colony pedigree collapse would have been significant given the relatively small number of settlers. And overall, it becomes more likely as the number of descendents grows, putting constant downward pressure on the numbers. Lastly I'd note that we also have to conveniently ignore the fact that over the centuries people move and not ALL Mayflower descendents are going to be American.

In all, we can't really be sure. One or two early families of unusually large size will greatly affect the estimates. But if I'm speculating, the General Society of Mayflower Descendants probably happily grabbed some estimates from studies showing historical fertility rates of 7-8 children per family, and probably generates some attention for itself claiming 1 in 10 Americans are related to Mayflower pilgrims.

Notes:

^[1] GSMD Annual Report 2017 - 2018

^[2] Gower, Nathan. BBC. Mayflower 400 years: How many people are related to the Mayflower pilgrims?

^[3] CDC: Mean Age of Mothers is on the Rise: United States, 2000–2014

^[4] Based on: 16 generations since the Mayflower arrived (25 years each), 20 million Mayflower descendents in the current generation (reduced from the 35 figure since more than one generation of people is alive at the same time), and 50 people in the initial generation (the Mayflower carried 102 passengers but half died in the first winter).

^[5] Hacker, J David. (2003). Rethinking the "Early" Decline of Marital Fertility in the United States. Demography, Vol. 40, No. 4

^[6] Bailey, Martha J. and Hershbein, Brad J. U.S. Fertility Rates and Childbearing, 1800 To 2010

^[7] Demos, John. (2000). A Little Commonwealth: Family Life in Plymouth Colony. Oxford University Press