The quasibinary system KBi2-Pb-x(x) has been investigated, both experimentally and theoretically. Phases with compositions 0 <= x <= 1.2 were synthesized and structurally characterized by X-ray diffraction experiments. For low values of x (0 <= x <= 0.6), KBi2-Pb-x(x) adopts the cubic Laves-phase structure MgCu2 (space group Fd3m), which contains a rigid framework of corner-condensed symmetry-equivalent tetrahedra formed by randomly distributed Bi and Pb atoms. For compositions x >= 0.6, these tetrahedra become alternately elongated and contracted. The distortion of the framework lowers the space-group symmetry to F43m (KBi1.2Pb0.8, F43m, Z = 8, a = 9.572(1) angstrom). Magnetometer measurements show that KBi2 (X = 0) is metallic and goes through a superconducting transition below 3.5 K. First principles calculations reveal that the Fd3m -> F43m distortion is largest for KBiPb (x = 1.0), which at the same time turns into a semiconductor. Thus, F43m KBiPb corresponds to a proper charge-balanced Zintl phase, K+[BiPb](-), with separated polyanionic tetrahedra, (Bi2Pb2)(2-). However, it was not possible to prepare F43m KBiPb. Syntheses attempting to increase the Pb content in KBi2-xPbx above x = 0.8 yielded additional, not yet characterized, ternary phases.