As far as I know, I'm 3/4 English and 1/4 Irish, by way of Arkansas and Kentucky. A lot of information is missing, so I'm basing that judgment on what their surnames sound like.
I wonder how many people can name all 8 of their great-grandparents, and say where they lived?
When people tell me, "My ancestors came from England" (or whatever) I am often tempted to ask, "Yeah, but where did they come from before THAT?"
People in this hemisphere tend to assume that the slate gets wiped clean when you cross an ocean; that if you know what country your ancestors embarked from, that's all there is to know about who you are. Why should they assume that?