@dulcinea: Yes, but in Germany we don't have gowns, caps etc., and the only degree associated with a special hat at all is a doctorate. And this hat is what I in English would call a "mortarboard", that is a cylindrical structure with a square on top and a tassel hanging down.
In our faculty, whenever someone finished his doctorate, the members of the workgroup would secretly make a cardboard mortarboard, the "Doktorhut", and decorate it with newspaper cut-outs, plasticine figures, small toys etc. to depict the candidate's weaknesses or hobbies (or, rarely, something to do with the thesis) and present it to the new doctor when he emerged from the dean's office after the final exam. I still have mine. But for the Diplom, there was no ceremony, no hat, etc. Even if Bachelor's or Master's degrees had been established at the time, I doubt whether anyone would have had a mortarboard then, and I'm sure they still don't. Show anyone in the street a mortarboard, and if they recognize the thingy at all, they'll call it a "Doktorhut".
A "Barett", to me, is a kind of soft cap either soldiers or Frenchmen wear :o) (beret)