Two independent data streams—the “zero-error stream” and the “rare-error stream”—are to be transmitted over a noisy discrete memoryless channel with feedback. Errors are tolerated only in the rare-error stream, provided that their probability tends to zero. Clearly the rate of the error-free stream cannot exceed the channel’s zero-error feedback capacity, and the sum of the streams’ rates cannot exceed the channel’s Shannon capacity. Using a suitable coding scheme, these necessary conditions are shown to characterize all the achievable rate pairs. Planning for the worst—as is needed to achieve zero-error communication—and planning for the true channel—as is needed to communicate near the Shannon limit—are thus not incompatible.