In traditional communication theory, it is assumed that error correcting codes may be decoded with algorithms that perform perfectly. Noise, however, provides a fundamental limit to computation systems just as it does to communication systems. In this work, we investigate the effect of noise in message-passing decoders for low-density parity-check codes. We observe that the concentration of the performance of the decoder around its average performance continues to hold when noise is introduced into message-passing. Given the concentration result, density evolution equations for a simple noisy message-passing decoder are derived. Analytic computation of thresholds shows that performance degrades smoothly as decoder noise increases. Decoding is robust to noise in the decoder.