The origin and evolution of gas in debris disks are still not well understood. Secondary gas production from cometary material or a primordial origin have been proposed. So far, observations have mostly concentrated on CO, with only a few C observations available. We overview the C and CO content of debris disk gas and test state-of-the-art models. We use new and archival Atacama Large Millimeter/submillimeter Array (ALMA) observations of CO and C i emission, complemented by C ii data from Herschel, for a sample of 14 debris disks. This expands the number of disks with ALMA measurements of both CO and C i by 10 disks. We present new detections of C i emission toward three disks: HD 21997, HD 121191, and HD 121617. We use a simple disk model to derive gas masses and column densities. We find that current state-of-the-art models of secondary gas production overpredict the C-0 content of debris disk gas. This does not rule out a secondary origin, but might indicate that the models require an additional C removal process. Alternatively, the gas might be produced in transient events rather than a steady-state collisional cascade. We also test a primordial gas origin by comparing our results to a simplified thermochemical model. This yields promising results, but more detailed work is required before a conclusion can be reached. Our work demonstrates that the combination of C and CO data is a powerful tool to advance our understanding of debris disk gas.