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.

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


  [1]: https://en.wikipedia.org/wiki/Project_Mercury#Selection_and_training
  [2]: https://en.wikipedia.org/wiki/Jim_Lovell
  [3]: https://en.wikipedia.org/wiki/Project_Gemini
  [4]: https://en.wikipedia.org/wiki/Apollo_program