A full quartic potential energy surface is determined for core-ionized methane and used to investigate coupling between vibrational modes. A strong Fermi resonance is found between the first excited state of the symmetric stretching mode nu(1)(') and a doubly excited bending mode, whereas the corresponding interaction is less pronounced for v(1)(')=2. In terms of the carbon 1s photoelectron spectrum of methane, the net effect of the mode coupling is to reduce the apparent contribution from anharmonicity to peak positions. The contribution from anharmonicity to the intensity of each peak is dominated by cubic and quartic terms in the symmetric stretching coordinate, and remains significant. This resolves a paradox pointed out in a recent experimental work [Carroll , Phys. Rev. A 59, 3386 (1999)].