Yes, cwelli, there is a connection, but its not the most straightforward. The common ancestor of both Amelia Sennett (born 28 Jul 1869, in the Stoke Damerel reg. district) and Richard Sennett (b1847) is William Sennett born 1768. William married three times. Amelia is descended from William and wife Mary Hitchens and Richard is descended from another of William's wives, Elizabeth Williams.
Richard b 1847 is the son of Richard b abt 1816 who married Ann Marrack, and gson of William and Elizabeth Williams. Amelia is the ggrandaughter of William through his wife Mary Hitchens. Amelia's father is William James, her grandfather is James b 1808 (who married Amelia Hitchens)and her paternal greatgrandparents are William Sennett and Elizabeth Williams. So Amelias father William James and Richard shared a common grandfather, but not grandmother. I haven't a clue what relation that makes them, but its definitely a relationship.
If you send me a PM with your email address, I'll print out a report that shows all these generations and how they link together. It will be a pdf, so I'll need to email it rather than attach it to a message here.
You were interested in how the misconception arose. I wonder if it perhaps was in the way the term "cousin" used to be used. We think of cousins as being first cousins, but in the past the term was often used a lot more loosely than that to refer to anyone who was related, even distantly. All these people did live in the same area, so it would have been quite likely that the families saw each other, perhaps without children understanding the exact relationship, which may then have been misinterpreted as being closer than it was when they grew up. I have an example in another tree of two men calling each other "cousins", but they don't even have a common greatgrandparent (the relationship is much more distant), let alone the common grandparent we would expect with first cousins.
