We discuss the connection of a violation of Bell's inequality and the non-Kolmogorovness of statistical data in the EPR-Bohm experiment. We emphasize that nonlocalty and "death of realism" are only sufficient, but not necessary conditions for non-Kolmogorovness. Other sufficient conditions for non-Kolmogorovness and, hence, a violation of Bell's inequality can be found. We find one important source of non-Kolmogorovness by analyzing the axiomatics of quantum mechanics. We pay attention to the postulate (due to von Neumann and Dirac) on simultaneous measurement of quantum observables given by commuting operators. This postulate is criticized as nonphysical. We propose a new interpretation of the Born-von Neumann-Dirac rule for the calculation of the joint probability distribution of such observables. A natural physical interpretation of the rule is provided by considering the conditional measurement scheme. We use this argument (i.e., the rejection of the postulate of simultaneous measurement) to provide a motivation for the non-Kolmogorovness of the probabilistic structure of the EPR-Bohm experiment. Copyright © 2011 American Scientific Publishers All rights reserved.