We use the Dirac equation in a fixed black hole background and different independent techniques to demonstrate the absence of fermionic bound states around a Schwarzschild black hole. In particular, we show that no embedded eigenvalues exist which has been claimed for the case when the energy is less than the particle’s mass. We explicitly prove that the claims regarding the embedded eigenvalues can be traced back to an oversimplified approximation in the calculation. We conclude that no bound states exist regardless of the value of the mass.