As Natalie says, we all have zillions of ancestors at that distance. Putting it another way, everybody alive in England at that time is an ancestor of all modern English and Anglo-American people, if they have any living descendants at all (many don't).
So in principle you can look at Henry I or anybody else from that time and assume that they're your ancestor.
But if you go round claiming to be descended from Henry I, people will assume you're talking about a traceable line, not just a statistical calculation.
Having a traceable line is just lucky. It's down to accidental survival of records. Doesn't stop people bragging about it.
Many people do have traceable lines. But a much bigger number think they do.
Usually in America, the brick wall is the immigrant, ifyou can get that far, because there's no clue where the immigrant came from. Generally there were many people of the same name in England. (They thought they couldn't go wrong with Aquila Chase, how many of those could there be, but they found the wrong one.)
In the 19th century it was very common for people to find the richest English person of the same name and claim that he was the immigrant. They would concoct stories about how the runaway son of an earl stowed away on the Mayflower and became a Connecticut carpenter. Or whatever.
Some of those people were deluded. Others thought it was just harmless family entertainment.
Often they didn't realize that the English person they were laying claim to had fake ancestry already. In England it had been common since the 16th century for families to come from nowhere, make a lot of money, and then have a fake pedigree done, showing their descent from a brave Norman knight who impressed the Conqueror. And for a fee they could have these claims enrolled in official records (which were only ever records of what people claimed).
In America, many family stories were republished in pretentious books with titles like Warren County Families. The compilers of those books just used what they were given without questioning or researching or checking anything.
Then came the LDS and the internet and now genealogy is awash with junk. You're bound to come across some, because some has been written about the Lyon family.