The recent measurement by ATLAS of light-by-light scattering in LHC Pb-Pb collisions is the first direct evidence for this basic process. We find that it excludes a range of the mass scale of a nonlinear Born-Infeld extension of QED that is ≲100 GeV, a much stronger constraint than those derived previously. In the case of a Born-Infeld extension of the standard model in which the U(1)_{Y} hypercharge gauge symmetry is realized nonlinearly, the limit on the corresponding mass reach is ∼90 GeV, which, in turn, imposes a lower limit of ≳11 TeV on the magnetic monopole mass in such a U(1)_{Y} Born-Infeld theory.