The Amazing Mumford lives in a castle next door to a castle owned by the Count. In "Who Stole the Count's Thunder?" from The Sesame Street Bedtime Storybook, it is implied that Mumford has recently moved in, as he comments that the walls are so thin in these newly built castles.
Living so close to the Count, Mumford has a hard time sleeping at night due to his vampire friend's late-night counting, which is followed by boisterous thunder. Mumford steals the Count's thunder, allowing him to sleep so that he can work from home the next day. He's so well rested that he is able to produce 712 rabbits from his hat from the comfort of home.