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According to the Wikipedia article on Project Mercury

Navy Lt (later Capt) Jim Lovell, who was later an astronaut in the Gemini and Apollo programs, did not pass the physical tests.

Lovell flew on 4 different missions, notably as the commander of the famous Apollo 13. He was also part of the crew for Gemini 7 (Pilot) & 12 (Command Pilot) and Appolo 8 (Command Module Pilot).

So it seems at least one of the candidates that failed the tests did later become an astronaut.