The problem of determining one-sided tolerance limits or, equivalently, one-sided confidence limits on a quantile of a normal population with variance σ2= σ2b+ σ2bis considered in this article. The tolerance limits are obtained using data from a one-way balanced randomeffects sample from this population. Two procedures are introduced—a simple approximate method based on an asymptotic expansion and a computer-intensive procedure that provides very nearly the nominal confidence in the presence of the nuisance variance-component ratio. For the computer-intensive procedure, tables are provided that eliminate the need for most of the computation for some important situations. Tolerance limits for strength are routinely used in structural design. Material strength, particularly for composite materials, often exhibits considerable between-batch variability. A tolerance limit that provides nearly the nominal confidence is usually preferable to a conservative limit, since a conservative tolerance limit underestimates the capability of the material. An aircraft-industry application to determining tolerance limits for composite material properties in the presence of between-batch variability is discussed.