Brinkman was born in Rotterdam, Netherlands, in 1902. He was the son of architect Michiel Brinkman (1873–1925), who established a firm in Rotterdam in 1910 and was known for designing the Spangen neighborhood of Rotterdam in 1922. Johannes studied civil engineering at the Delft University of Technology (Dutch: Technische Universiteit Delft). After Michiel's death in 1925, Johannes took charge of the father's architectural firm and entered into a partnership with architect Leendert van der Vlugt. The results of that collaboration include the Van Nelle Factory and the Feijenoord Stadion.
After the death of Van der Vlugt in 1936, Brinkman teamed up with architect Johannes Hendrik van den Broek. The firm's work during this time including a new Rotterdam Cruise Terminal for the Holland-America Line.
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