Certainly possible. My sister has a similar theory. My grandmother Sarah was a Carey, Descended from John -> Isiah Wilson.
We know John was born in Ireland, but unfortunately at least two different John Carey's migrated from Ireland to the US around the same time, and it's not like John Carey is a rare name in Ireland. There is a John Carey descended from William, but at this point no proof that was one of the John Carey's that went to the US, or if it was the one from which I descended.
There are numerous generations between William Carey and the hordes of Careys that migrated to the US during the potato famines so if you can trace your Carey ancestor's back to Ireland the odds aren't bad.