I too have an Oliver, but he is rarely called Ollie. His nickname is the Schnizzler. It was a name I had liked for years and considered it for one of my two human boys. The child that ended up not being called Oliver is Elliot, so more often than not, I call Oliver when I mean Elliot and vice versa.
Henry's name was chosen after much discussion on here, as well as within the family. My son Elliot came up with Henry and it just seemed to suit him. It had to "match" the sound of Oliver so that as Kristen mentioned, when calling out in public it went well together. He is often called "Hanky Panky" these days as he can be a bit of a devil.
I did the same thing with my human children as well. The older one is Benjamin, or Ben which goes well with Elliot, and I think all four of my boys suit their names.