We have measured resonance strengths and energies for dielectronic recombination (DR) of Mg-like Fe XV forming Al-like Fe XIV via N = 3 → N' = 3 core excitations in the electron-ion collision energy range 0-45 eV. All measurements were carried out using the heavy-ion test storage ring at the Max Planck Institute for Nuclear Physics in Heidelberg, Germany. We have also carried out new multiconfiguration Breit-Pauli (MCBP) calculations using the AUTOSTRUCTURE code. For electron-ion collision energies ≲ 25 eV we find poor agreement between our experimental and theoretical resonance energies and strengths. From 25 to 42 eV we find good agreement between the two for resonance energies. But in this energy range the theoretical resonance strengths are ≈31% larger than the experimental results. This is larger than our estimated total experimental uncertainty in this energy range of ±26% (at a 90% confidence level). Above 42 eV the difference in the shape between the calculated and measured 3s3p(1P1)nl DR series limit we attribute partly to the nl dependence of the detection probabilities of high Rydberg states in the experiment. We have used our measurements, supplemented by our AUTOSTRUCTURE calculations, to produce a Maxwellian-averaged 3 → 3 DR rate coefficient for Fe XV forming Fe XIV. The resulting rate coefficient is estimated to be accurate to better than ±29% (at a 90% confidence level) for kBTe ≥ 1 eV. At temperatures of kBTe ≈ 2.5-15 eV, where Fe XV is predicted to form in photoionized plasmas, significant discrepancies are found between our experimentally derived rate coefficient and previously published theoretical results. Our new MCBP plasma rate coefficient is 19%-28% smaller than our experimental results over this temperature range.