The molecular makeup of the offspring of a dividing cell gradually becomes phenotypically decorrelated from the parent cell by noise and regulatory mechanisms that amplify phenotypic heterogeneity. Such regulatory mechanisms form networks that contain thresholds between phenotypes. Populations of cells can be poised near the threshold so that a subset of the population probabilistically undergoes the phenotypic transition. We sought to characterize the diversity of bacterial populations around a growth-modulating threshold via analysis of the effect of non-genetic inheritance, similar to conditions that create antibiotic-tolerant persister cells and other examples of bet hedging. Using simulations and experimental lineage data in Escherichia coli, we present evidence that regulation of growth amplifies the dependence of growth arrest on cellular lineage, causing clusters of related cells undergo growth arrest in certain conditions. Our simulations predict that lineage correlations and the sensitivity of growth to changes in toxin levels coincide in a critical regime. Below the critical regime, the sizes of related growth arrested clusters are distributed exponentially, while in the critical regime clusters sizes are more likely to become large. Furthermore, phenotypic diversity can be nearly as high as possible near the critical regime, but for most parameter values it falls far below the theoretical limit. We conclude that lineage information is indispensable for understanding regulation of cellular growth.