The Ostwald law of dilution provides a satisfactory description of the concentration dependence of the conductivity of weak electrolytes like CHCOOH and NHOH.  The variation of molar conductivity is essentially due to the incomplete dissociation of weak electrolytes into ions. 
For strong electrolytes, however, Lewis and Randall recognized that the law fails badly since the supposed equilibrium constant is actually far from constant. This is because the dissociation of strong electrolytes into ions is essentially complete below a concentration threshold value. The decrease in molar conductivity as a function of concentration is actually due to attraction between ions of opposite charge as expressed in the Debye-Hückel-Onsager equation and later revisions.
Even for weak electrolytes the equation is not exact. Chemical thermodynamics shows that the true equilibrium constant is a ratio of thermodynamic activities, and that each concentration must be multiplied by an activity coefficient. This correction is important for ionic solutions due to the strong forces between ionic charges. An estimate of their values is given by the Debye–Hückel theory at low concentrations.