He attended the secondary school in the town and then studied stage plays in Ljubljana under Vida Juvan. He finished his studies in 1961.
He first appeared in the film Moments of Decision (Trenutki odločitve. 1955), directed by František Čap. Later, he worked in the Ljubljana Drama Theatre, where he was also the director from 1977 until 1981.
He was a professor of theatre at the Ljubljana Academy of Theatre, Radio, Film and Television.
He died aged 79 in Ljubljana.
Foreign his work, he was awarded the Župančič Award (1974), the award for cultural achievements in the city of Ljubljana, the Prešeren Award (1977 and 1985), the highest Slovenian award in the field of culture, the Borštnik Ring (1984), the highest Slovenian award for actors, the Silver Order of Freedom of Republic of Slovenia (1996), and the Sterija Prize in Novi Sad in 1984. At his death, Bibič was described by other Slovenian actors and writers as a giant, the prince of the theatre, and a legendary acting folk character. A deep admiration of him and appreciation of his work was also expressed by the President of Slovenia Danilo Türk and the Prime Minister Janez Janša.