Purpose - For the considered system, an enumeration method is applicable to evaluate the exact system reliability only for very small-sized systems, because, when the size of system is large, it takes huge execution time. Therefore, the paper provides approximate values for the system reliability as useful for calculating the reliability of large systems in a reasonable execution time. Design/methodology/approach - The paper provides upper and lower bounds of the system reliability, and limit theorem for the reliability of our considered system in i.i.d. case. Findings - The paper experimentally finds that the proposed upper and lower bounds are effective when component reliabilities close to one or the value of k becomes larger. Next, it concludes approximate values for approximate equation derived from the limit theorem are always smaller than lower bound through numerical experiments. Research limitations/implications - The upper and lower bounds for the reliability of a system can be calculated by using the reliability of a small system by the same idea as previous study for two-dimensional system. Practical implications - Up to now some researchers studied multi-dimensional consecutive-k-out-of-n:F systems, and showed promising applications of such multi-dimensional models, e.g. diagnosis of a disease diagnosed by reading an X-ray. As another examples, three-dimensional system can be applied for the mathematical model of a three-dimensional flash memory cell failure model, and so on. Originality/value - The paper considers a kind of three-dimensional k-within-consecutive-r-out-of-n:F system. It proposes upper and lower bounds of the system reliability and limit theorem.