There were a few callers here in Missouri who I've heard could fiddle while they called squares. That was several years ago, though. At this point (around the St. Louis area anyway) there's only one or maybe two that I know of who can call a dance while they play.
A lot more common are callers who can dance while they call, like Jim Thaxter from the Columbia, MO area and one of my favorite callers.
Most of the callers I know have to read the dances off of little cards or their IPad, so would be of little use playing an instrument or dancing while they call. My guess is that they are familiar with a couple of hundred contras and squares, but that's too many to memorize.