Due to structural and functional abnormalities or genetic variations and mutations, there may be dysfunctional molecules within an intracellular signaling network that do not allow the network to correctly regulate its output molecules, such as transcription factors. This disruption in signaling interrupts normal cellular functions and may eventually develop some pathological conditions. In this paper, computation capacity of signaling networks is introduced as a fundamental limit on signaling capability and performance of such networks. The computation capacity measures the maximum number of computable inputs, that is, the maximum number of input values for which the correct functional output values can be recovered from the erroneous network outputs, when the network contains some dysfunctional molecules. This contrasts with the conventional communication capacity that measures instead the maximum number of input values that can be correctly distinguished based on the erroneous network outputs. The computation capacity is higher than the communication capacity, if the network response function is not a one-to-one function of the input signals. By explicitly incorporating the effect of signaling errors that result in the network dysfunction, the computation capacity provides more information about the network and its malfunction. Two examples of signaling networks are studied here, one regulating caspase3 and another regulating NFkB, for which computation and communication capacities are analyzed. Higher computation capacities are observed for both networks. One biological implication of this finding is that signaling networks may have more capacity than that specified by the conventional communication capacity metric. The effect of feedback is also studied. In summary, this paper reports findings on a new fundamental feature of the signaling capability of cell signaling networks.