The pressure-induced changes in the electronic band structures of In-containing nitride alloys, InxGa1-xN and InxAl1-xN are examined experimentally as well as by ab initio calculations. It is found that the band gap pressure coefficients, dE(g)/dp, exhibit very large bowing with x, and calculations with simulation of clustered distributions of the In atoms over the cation sites show a strong enhancement of this effect. This relates well to the experimental data obtained from pressure dependent photoluminescence measurements for InxGa1-xN and InxAl1-xN layers, performed in this work, and combined with existing data for InxGa1-xN layers. We discuss possible explanations of the anomalously large magnitude of the dE(g)/dp bowing in these nitride alloys.