In unreplicated two-way factorial designs, it is typical to assume no interaction between two factors. However, violations of this additivity assumption have often been found in applications, and tests for non-additivity have been a recurring topic since Tukey's one-degree of freedom test (Tukey, 1949). In the context of randomized complete block designs, recent work by Franck et al. (2013) is based on an intuitive model with "hidden additivity," a type of non-additivity where unobserved groups of blocks exist such that treatment and block effects are additive within groups, but treatment effects may be different across groups. Their proposed test statistic for detecting hidden additivity is called the "all-conguration maximum interaction F-statistic" (ACMIF). The computations of the ACMIF also result in a clustering method for blocks related to the k-means procedure. When hidden additivity is detected, a new method is proposed here for condence intervals of contrasts within groups that takes into account the error due to clustering by forming the union of standard intervals over a subset of likely congurations.