Yes, that's what I was thinking and why I am hesitant to think this is them. My Washigntons did that - some brothers emigrated from Littleport in Cambrideshire to Dunkirk, then some more kids and the father a few years later, then the father goes back and brings the rest of the family a year after that. I have considered myself lucky to have foudn William's name, as we did not know it previously. Unfortunately this is my grandmother's line and she and my grandfather divorced around 1950 and he never spoke of her, she lived very far away and was not in touch, and of course now anyone who knew anything has passed. I did know he was from Stratford and then I found the obituary naming his children so I definitely found him. I suspect I may never be able to confirm anything further back. But I am very inexperienced at this, so I hope perhaps one day I will find something more conclusive!
You have given me an idea, however. There are only 2 families with the last name of Gibbs who have members born in England and living anywhere within all of Chautauqua County, New York on the 1860 US Federal Census. One of them has the William that seems to possibly be mine. Perhaps my next step will be to do some research on the other Gibbs family living in the same town but born in England and see if they come from Stratford-Upon-Avon!
Ok, I will not take up any more of your time except to express my many thanks and gratitude for your help and suggestions!