Blends of dimethyl ether and methane have been studied in many different settings; however, the near-limit flame behavior of these mixtures is experimentally underexplored. This study examines the extinction behavior and flame structure of dimethyl ether/methane diffusion flames. An ozone-assisted counterflow burner is utilized to investigate both hot flames and cool flames. Methane addition decreases the hot flame extinction limit on a molar basis but promotes it on a mass basis. However, for cool flames methane is extremely inhibitive, reducing the cool flame extinction limit even when it replaces inert nitrogen due to its scavenging of the radicals needed for low-temperature branching. The cool flame structure is seen to exhibit substantial reactant leakage, which is not well captured by a state-of-the-art kinetic model. Modifications to the kinetic model show that updating the fuel + hydroxyl radical reaction and the hydroperoxyalkyl decomposition reaction significantly improve the cool flame extinction limit predictions while simultaneously maintaining fidelity to previous validations of homogeneous reactor results. This study highlights the value of the counterflow cool flame platform for chemical kinetic model validation at low temperatures.