Not if you have a Dremel, or similar. Mine is a Black & Decker clone..
I just polished a formally brushed J le C Kompressor case and bezel. I removed the movement and the three crowns, masked the crystal with parcel tape (probably not necessary, but better safe that sorry) then set to with the Cape Cod. About 25 minutes later, the job was done. Easy!
My method was to cut a half-inch strip of the CC cloth, fold it like a concertina, make a hole through the centre of the wad, then mount it on an arbour. Making a hole through the centre of the wad was probably the hardest part of the operation. CC is surprisingly tough stuff.
Running the clone Dremel at low speed worked wonders. The result... quite quickly achieved, was a perfectly polished case and bezel. It has transformed the watch.