In photoionized gases with cosmic abundances, dielectronic recombination (DR) proceeds primarily via nlj → nl ′ j ′ core excitations (∆n = 0 DR). We have measured the resonance strengths and energies for Fe XVIII to Fe XVII and Fe XIX to Fe XVIII ∆n = 0 DR. Using our measurements, we have calculated the Fe XVIII and Fe XIX ∆n = 0 DR rate coefficients. Significant discrepancies exist between our inferred rates and those of published calculations. These calculations overestimate the DR rates by factors of ∼ 2 or underestimate it by factors of ∼ 2 to orders of magnitude, but none are in good agreement with our results. Almost all published DR rates for modeling cosmic plasmas are computed using the same theoretical techniques as the above-mentioned calculations. Hence, our measurements call into question all theoretical ∆n = 0 DR rates used for ionization balance calculations of cosmic plasmas. At temperatures where the Fe XVIII and Fe XIX fractional abundances are predicted to peak in photoionized gases of cosmic abundances, the theoretical rates underestimate the Fe -2 -XVIII DR rate by a factor of ∼ 2 and overestimate the Fe XIX DR rate by a factor of ∼ 1.6. We have carried out new multiconfiguration Dirac-Fock and multiconfiguration Breit-Pauli calculations which agree with our measured resonance strengths and rate coefficients to within typically better than ∼ < 30%. We provide a fit to our inferred rate coefficients for use in plasma modeling. Using our DR measurements, we infer a factor of ∼ 2 error in the Fe XX through Fe XXIV ∆n = 0 DR rates. We investigate the effects of this estimated error for the well-known thermal instability of photoionized gas. We find that errors in these rates cannot remove the instability, but they do dramatically affect the range in parameter space over which it forms.