The photoluminescence spectra from a quantum-dot exciton weakly-coupled to a planar photonic-crystal cavity is experimentally investigated by temperature tuning. Significant resonance shifts of the cavity mode are observed as the cavity mode spectrally approaches that of the exciton mode, showing the appearance of cavity-to-exciton attraction or mode pulling. Cavity-mode spectral shifts are also found theoretically using a master equation model that includes incoherent pump processes for the coupled exciton and cavity, pure dephasing, and allows for photon emission via radiation modes and the leaky cavity mode. Both experiments and theory show clear cavity mode spectral shifts in the photoluminescence spectra, when certain coupling parameters are met. However, discrepancies between the experimental data and theory, including more pronounced spectral shifts in the measurements, indicate that other unknown mode-pulling effects may also be occurring.