Reliable evaluation of a product's quality assurance capability is difficult to quantify due to the intricacies of the quality assurance process and the uncertain relationships between the product quality and its influence factors. To address this problem, with the aid of information-poor theory, a new concept, quality-achieving reliability, is proposed, and a corresponding evaluation model is structured to calculate the probability of achieving product quality as a function of the influence factors. Using the decomposition and composition method, quality classifications of the product evaluation indicator and its influence factors are produced to extract the cumulative distribution of the product quality and its elements. Using the grey relational grade method, a weight analysis of the impacts of the influence factors on the product quality is performed to obtain the status composite values of the influence factors. Under the condition of poor information and relying on the status composite values based on the cumulative distribution, a quality-achieving reliability function is established to reflect the quality assurance capability and quality control level of the product and its elements. An experimental investigation of the vibration acceleration of tapered roller bearings demonstrates the application of the proposed model.