The nuts themselves are the same size or very close to the washer so in most cases no. On some of the older engines the crank shaft threads were cut shorter and the washer was needed to make up the space.
I just posted this photo of three different typre of nuts. The two nut shaped are Higgly's and the brass one is used for planes that are a little tail heavy instead of adding lead.
These are very cheap compared to a good spinner and don't break like the cheap plastic spinners.