Predictions of future atmospheric CO2 levels and their impacts depend on understanding the role of the terrestrial biosphere in the carbon cycle. Major efforts have been focused on improving ecosystem models, which are numerical representations of a collection of biogeochemical processes, to improve model performance and generate more realistic predictions. However, despite notable research, model predictions continue to diverge. Here, we consider this issue from an information-theoretic perspective. We choose to measure the complexity of net ecosystem exchange (NEE), an important flux in the global C cycle, and evaluate a "simple" case: how NEE switches between a source (positive values) and sink (negative values) of atmospheric CO2 over time, at half-hourly and daily time scales. Instead of assuming specific model formulations for these source-sink dynamics, we let each observed or simulated time series of NEE at 42 sites from North American Carbon Program Site-level Interim Synthesis to inform how NEE is best represented, with the smallest optimal predictors: a type of hidden Markov model called the epsilon machine (eM). The eM internally stores information (measured by its statistical complexity) that must be "remembered" to optimally predict future source-sink dynamics. Here we show, at the half-hourly scale, eMs inferred from model-simulated NEE have much lower statistical complexity than those inferred from flux observations, suggesting that simulated NEE source-sink dynamics are less complex and potentially more instantaneous than those of observations. This model-data mismatch is only marginally explained by differences in conventional information measures of entropy rate (i.e., the apparent randomness of source-sink dynamics) and excess entropy (i.e., future source-sink dynamics directly predictable from past states). Such mismatch is not evident at the daily scale. Second, we show that when a site is disturbed, its source-sink dynamics generally decrease in complexity, despite the relatively small change in entropy rate. In summary, these information measures of land-carbon dynamics can provide insights on model adequacy across scales, effects of ecological memory and legacies, and the overall predictability of biosphere C fluxes.