According to the [Wikipedia article][1] on Project Mercury

> Navy Lt (later Capt) [Jim Lovell][2], who was later an astronaut in the [Gemini][3] and [Apollo programs][4], did not pass the physical tests.

Lovell flew on 4 different missions, notably as the commander of the famous [Apollo 13][5]. 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.