The influence of hydrostatic pressure on the emission and absorption spectra measured for various types of InGaN structures (epilayers, quantum wells, and quantum dots) is studied. While the known pressure coefficients of the GaN and InN band gaps are about 40 and 25 meV/GPa, respectively, the observed pressure-induced shifts in light emission energy in the InGaN alloys differ significantly from concentration-interpolated values. With increasing In concentration, and thus decreasing emission energy, the observed pressure coefficients become very small, reaching zero for emission energies similar to2 eV (roughly the value of the InN band gap). On the other hand, the pressure coefficient derived from absorption experiments exhibit a much smaller decrease with decreasing energy when referred to the same scale as the emission data. First-principles calculations of InGaN band structures and their modification with pressure are performed. The results are not able to explain the huge effect observed in the emission experiments, but they are in good agreement with the optical absorption data. Significant bowings of the band gap and its pressure coefficients are found, and they are especially large for small In concentrations. This behavior is related to the changes in the upper valence band states due to In alloying. Some possible mechanisms are discussed which might be expected to account for the low pressure coefficients of the light emission energy and the difference between the sensitivity of the emission and absorption to pressure.