In 1962 the right-handed Bungert reached the quarterfinals of the International Australian Championships, the doubles finals of the International French Championships and the International Tennis Tournament of Monte Carlo.
Bungert was ranked as high as World No. 4 for 1964 by Lance Tingay of The Daily Telegraph.
After reaching the semifinals in 1963 (beating Roy Emerson before losing to Chuck McKinley) and 1964 (losing to Emerson), the unseeded Bungert was the second German player (thirty years after Gottfried von Cramm) to reach the Wimbledon men’s finals in 1967 when he beat Roger Taylor in five sets.
However, he lost the final in straight sets against the Australian John Newcombe.
In 1970 he (with Christian Kuhnke) was part of the German Davis Cup team which lost the finals against the U.S. 0:5. In July of the same year he won his only career singles title in Düsseldorf.
In the eighties Bungert was captain of the German Davis Cup team (Boris Becker and Michael Westphal) which lost the finals against Sweden 2:3. Nikola Pilić became his successor as captain.
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