They had to go back to the 1600's to get that, making the connections, even if true, basically meaningless.
Go back a thousand years, and your family tree includes literally every white person, assuming you're white. So I'm related to everyone on that chart too, probably don't have to go back a 1000 years to find a common English ancestor.
http://www.livescience.com/29390-all-europeans-share-recent-ancestors.html
The daily mail article is kind of terribly written, but is relevant because half the famous people on this tree are presidents.
Lets do some rough math. You have 2 parents, 4 grandparents, 8 great grandparents. From 2015 to 1635 is 380 yrs. Lets say an average of 25yrs for each generation, that's 15 generations, 2^15 ancestors. That's 32,768 great great.... grandparents.
The reverse also applies, though not as neatly. If, on average, each person had two kids who survived to reproduce, any random person from the year 1635 would have about 33,000 descendants in the 'current generation'. Intermarriage would reduce this number, but population growth shows that shows our 2 kids estimate is low, increasing the total. Not being super scientific here, just looking for a ballpark estimate.
They're not looking at just the current generation, they're looking at everyone in that descending tree, ie, 33,000 people today, their 16,000 parents on that side of the family, etc, for a total of about (2^15+2^14+2^13...) = over 65,000 people.
So, any one random person from 1635 could reasonably have 65,000 people, living or dead, descended from them. Pulling 8 famous people out of a pool of 65,000 isn't totally unexpected.
EDIT:
/u/Searocksandtrees posted a link that's better than my stuff.
"Related" is really stretching the term if you have to go back to at least a 13x great-grandparent in the mid 1600s to get a link: basically your girlfriend is, what, Bush's 15th cousin once or twice removed? Not close enough to get invited to the Bush family picnic.
Check out this recent post regarding an article on relatedness of US presidents
Only a family tree formally made up can tell. After all, Churchill's mother was one of the Buccaneers, daughters of American New Wealth whose mothers took them to London in hopes of landing a lord (see To Marry an English Lord for how this worked).
If they are Mayflower, they might be related to the Adams, which gives you two presidents.
To say it's statistically improbable is erroneous, because as soon as you start connecting to the small upper class it becomes very likely you can trace a relationship to many more members of the group. We aren't marbles in a bucket, where only a few are cat's-eyes (famous), all statistically equal. We have societal groupings.
The trick is finding out if she actually is connected.
You've seen ads for "get your family arms"? That there is a family named Addison with arms doesn't mean all Addisons have arms. You have to trace descent. So a lot of people will skip the hard work and grab the ego stroke of notable people by a family name likeness via a female line often. Your gf's genitor may have been told all this by his great-aunt the sloppy genealogist, and thinks it true.
The trick is establishing the links. Unless you can prove descent from the land-owning classes, it's extremely unusual to be able to name an ancestor much before 1700 - more likely before 1840. Most US immigrants were not from such classes. Otherwise why would they have emigrated? Highland Scots could reel off the names of their forebears for generation after generation because their society was cemented by kinship connections. This was irrelevant once they had left Scotland and the links were forgotten.
The family tree itself is reasonable. As /u/Zither13 said, when it comes to the upper classes, the pool of acceptable spouses was smaller and it's more likely that you'll see famous upper class people related to each other. I've checked out several generations of it and it appears to be based on legitimate information. While there isn't always good genealogical information out there for the layman to find, during the late 19th/early 20th centuries, many people researched church records of births, marriages, and deaths in order to publish genealogical references of certain families going back to the 17th century, so it's certainly not out of the question for a lineage to be traceable that far. It's true that an individual today might not be able to name an ancestor from that period, but that doesn't mean the information isn't available with research.
Being descended from people who came over on the Mayflower has become a trope that is commonly associated with wealthy WASPs ... who are the end points of the family tree here. However, not all Mayflower descendants are wealthy WASPs. Through the 17th and 18th centuries in America, the birth rate increased and infant mortality decreased - population growth far exceeded simple replacement. Larger generations result in a larger chance of being descended from a particular family.
So your girlfriend's biological father may fit into the tree. However, does he? Is he assuming he does because his surname is Howland? Or has he documented it in some way? This isn't a question that can be answered here without more information (which you may not want to give out for obvious reasons).