It's both, Part. Borsalino Hatters got started with that style of hat in Allesandria, Italy in 1857. The firm's still in business, I think, but the style has become generic for all makers' hats of that general style, over the years. Fabrizio wears one at Michael and Apollonia's wedding. Willy Cicci wears one while controlling the crowd when Sonny beats up Carlo. A well-regarded gangster movie, starring Jeal Claude Belmondo and Alain Delon, was called "Borsalino," after their hats. Orthodox Jews wear them today--it's as much a part of their regular dress as a black suit and a yarmulke. The Borsalino firm also makes many other styles of hats and caps--it's a big producer.