We present a study of Langmuir isotherms and 2D bulk moduli of binary lipid mixtures, where changes in monolayer collapse pressure (Pi(c)) are followed while varying the relative amounts of the two components. For monolayers containing dipalmitoylphosphocholine (DPPC) with either hexadecanol (HD) or palmitic acid (PA), a distinctly non-monotonic change in Pi(c) is observed with varying composition. At low mole fractions, there is a slight decrease in Pi(c) as films get richer in DPPC, while a sharp increase to pure DPPC-like values is observed when the mole fraction exceeds similar to 0.7. The sudden transition in collapse pressure is explained using the principles of rigidity percolation, and important ramifications of this phenomenon for biological surfactant are discussed.