Cochems was born in 1877 at Sturgeon Bay, the county seat of Door County on Wisconsin's Door Peninsula. He was one of 11 children, and "the smallest of seven brothers." His older brother, Henry Cochems, preceded him at the University of Wisconsin-Madison, where he was a star football player and shotput thrower. Cochems also had a twin brother, Carl Cochems (1877–1954), who became a noted opera singer.