Coherent anti-Stokes Raman scattering (CARS) is used to measure relations between the resonant (Raman) and nonresonant (Kerr-type) optical nonlinearities of air-guided modes in a hollow-core photonic-crystal fiber (PCF). We demonstrate that, due to its interference nature, CARS provides a convenient tool for measuring the contribution of the fiber cladding to the total nonlinearity sensed by air-guided modes in hollow PCFs. On a Raman resonance with molecular vibrations in the gas that fills the fiber core, a two-color laser field is shown to induce optical nonlinearities that are several orders of magnitude higher than the nonresonant Kerr-type nonlinearities typical of air-guided PCF modes.