Characterizing the processes that drive microbial community assembly remains a key challenge in ecology. Several recent studies have argued that pairwise interactions may be insufficient to explain co-occurrence patterns in complex microbial communities, but most such studies have focused on synthetic communities not found in nature or microbes grown in contexts that differ widely from their natural environment. Moreover, most models of pairwise interactions rely on equilibrium assumptions that are not relevant to all natural communities, such as gut microbiomes or species in fluctuating environments. Microbes from fermented foods can be cultured in their natural abiotic environment in the lab and typically have moderate species complexity, making them tractable systems for studying community assembly processes. In this study, we investigated whether multi-species co-occurrence patterns can be predicted from pairwise interactions for microbes isolated from sourdough starters. Interaction parameters inferred from pairwise growth trajectories were suggestive of widespread coexistence between pairs of microbes in our species' pool. In communities of up to nine species, most species' presence and relative abundance could be reliably predicted based on a model of pairwise interactions. The inclusion of nonequilibrium demography in our model further improved the accuracy of our pairwise model. Our work contributes to the broader debate on the processes underlying community assembly by showing that pairwise interactions are predictive of community structure in a system of moderate species complexity.