Over the years, I've sold a few vintage coats with this unusual sleeve shape. Wide at the upper arm, the sleeve narrows into a tight fit at the cuff. This sleeve shape is often a bit longer than normal, and I've never been quite sure how they were supposed to look.
Mystery solved! Here are several images from 1955 that show the sleeve as it is meant to be worn. The cuff is pushed up to the elbow and is cut narrow so that it stays in place. This causes a pretty bunched look to the upper sleeve, while at the same time exposing your gloves or bracelets.