Bubník was born in Kelč, Czechoslovakia, and played in the Czechoslovak Extraliga.

He was inducted into the International Ice Hockey Federation Hall of Fame in 1997.

[1] He was tied with Canada's Harry Watson and Soviet Union's Valeri Kharlamov for the all-time Olympic scoring lead, until he was surpassed by Finland's Teemu Selänne in the 2010 Winter Olympics During his football career he played for RH Brno.

Over nine seasons in the Czechoslovak First League, he made 103 appearances, scoring 32 goals.

He earned 11 caps and scored 4 goals for the Czechoslovakia national football team from 1957 to 1960, and participated in the 1960 European Nations' Cup, scoring in the third place playoff match, in which Czechoslovakia beat France 2–0.