The disequilibrium combination of abundant methane and carbon dioxide has been proposed as a promising exoplanet biosignature that is readily detectable with upcoming telescopes such as the James Webb Space Telescope. However, few studies have explored the possibility of non-biological CH4 and CO2 and related contextual clues. Here, we investigate whether magmatic volcanic outgassing on terrestrial planets can produce atmospheric CH4 and CO2 with a thermodynamic model. Our model suggests that volcanoes are unlikely to produce CH4 fluxes comparable to biological fluxes. Improbable cases where volcanoes produce biological amounts of CH4 also produce ample carbon monoxide. We show, using a photochemical model, that high abiotic CH4 abundances produced by volcanoes would be accompanied by high CO abundances, which could be a detectable false positive diagnostic. Overall, when considering known mechanisms for generating abiotic CH4 on terrestrial planets, we conclude that observations of atmospheric CH4 with CO2 are difficult to explain without the presence of biology when the CH4 abundance implies a surface flux comparable to modern Earth's biological CH4 flux. A small or negligible CO abundance strengthens the CH4+CO2 biosignature because life readily consumes atmospheric CO, while reducing volcanic gases likely cause CO to build up in a planet's atmosphere. Furthermore, the difficulty of volcanically-generated CH4-rich atmospheres suitable for an origin of life may favor alternatives such as impact-induced reducing atmospheres.