The Born-Infeld form of the hydrogen atom has a spectrum that can be used to determine the physical viability of the theory, and place an experimentally relevant bound on the single parameter found in it. We compute this spectrum using the relativistic Dirac equation, and a form of the Born-Infeld potential that approximates the self-field corrections of the electron. Using these together, we can establish that if the Born-Infeld nonlinear electrodynamics is to be physically relevant, it must contain a fundamental constant that is well-below the original value proposed by Born. This work extends the original Schrödinger spectrum from [1] for the self-field correction, and shows that using the Dirac equation introduces minor corrections -but also gives access to a range for the fundamental constant that is below that attainable from non-relativistic considerations.