We present a comparison of existing experimental data for the radiative leptonic decays P → ν γ, where P = K or π and = e or µ, from the KLOE, PIBETA, E787, ISTRA+ and OKA collaborations with theoretical predictions based on the recent non-perturbative determinations of the structure-dependent vector and axial-vector form factors, F V and F A respectively. These were obtained using lattice QCD+QED simulations at order O(α em ) in the electromagnetic coupling. We find good agreement with the KLOE data on K → eν e γ decays from which the form factor F + = F V + F A can be determined. For K → µν µ γ decays we observe differences of up to 3 -4 standard deviations at large photon energies between the theoretical predictions and the data from the E787, ISTRA+ and OKA experiments and similar discrepancies in some kinematical regions with the PIBETA experiment on radiative pion decays. A global study of all the kaon-decay data within the Standard Model results in a poor fit, largely because at large photon energies the KLOE and E787 data cannot be reproduced simultaneously in terms of the same form factor F + . The discrepancy between the theoretical and experimental values of the form factor F − = F V − F A is even more pronounced. These observations motivate future improvements of both the theoretical and experimental determinations of the structure-dependent form factors F + and F − , as well as further theoretical investigations of models of "new physics" which might for example, include possible flavor changing interactions beyond V − A and/or non-universal corrections to the lepton couplings.