  We present the first nonperturbative numerical calculations of the
nonrelativistic hydrogen spectrum as predicted by first-quantized
electrodynamics with nonlinear Maxwell-Born-Infeld field equations. We also
show rigorous upper and lower bounds on the ground state.
  When judged against empirical data our results significantly restrict the
range of viable values of the new electromagnetic constant which is introduced
by the Born-Infeld theory.
  We assess Born's own proposal for the value of his constant.
