Bergmann was born into a Jewish family of Max Bergmann, a biochemistry professor, and Emmy Bergmann, a pediatrician in Berlin. His father would later be a professor of chemistry at the Rockefeller Institute for Medical Research. He began college in 1931, at the age of 16, at Technische Hochschule (now TU Dresden) under the mentorship of Harry Dember. Bergmann obtained his PhD at the age of 21 from the German University in Prague in 1936 under the direction of Philipp Frank.