We report systematic, high-precision measurements of the low-T (down to 70 mK) thermal conductivity kappa of YBa2Cu3Oy, La2-xSrxCuO4, and Bi2Sr2CaCu2O8+delta. Careful examinations of the Zn- and hole-doping dependences of the residual thermal conductivity kappa(0)/T, as well as the in-plane anisotropy of kappa(0)/T in Bi2Sr2CaCu2O8+delta, indicate a breakdown of the universal thermal conductivity, a notable theoretical prediction for d-wave superconductors. Our results point to an important role of electronic inhomogeneities, which are not considered in the standard perturbation theory for thermal conductivity, in the underdoped to optimally doped regime.