Grain futures contracts that permit physical delivery do so through an exchange of delivery instruments. Because delivery instruments can be held indefinitely, extant research shows that futures contracts that assign inflexible and low storage rates relative to the market price of storage facilitate basis nonconvergence. In response to the notable episode of non-convergence in the mid- to late-2000s, the Chicago Mercantile Exchange (CME) introduced variable storage rate (VSR) policies in the soft red winter (SRW) wheat and hard red winter wheat markets. The VSR mechanism functions by adjusting the storage rate to the price spread between sequential futures contract deliveries in the period right before the expiration of the nearby contract. In contrast, CME did not introduce a VSR to corn and soybean markets but chose to increase their fixed storage fees in 2008 and later in 2020. We study convergence performance for each of these markets from 2006-2020 and use time series techniques to show that flexible storage fee policies like the VSR reduce the magnitude and therefore the expected duration of nonconvergence in wheat markets. On the other hand, we do not find evidence that CME's higher fixed storage rates likewise reduce the expected duration of nonconvergence episodes in corn and soybeans markets - although perhaps not enough time has passed to evaluate the effectiveness of the most recent changes - or that index trader activity causes basis nonconvergence.