We present the results of an exhaustive study of the ionized gas in NGC 588, a giant Hii region in the nearby spiral galaxy M 33. This analysis uses a high number of diagnostics in the optical and infrared ranges. Four temperature diagnostics obtained with optical lines agree with a gas temperature of 11 000 K, while the [Oiii] lambda5007/lambda88 mum ratio yields a much lower temperature of ≈8000 K. This discrepancy suggests the presence of large temperature inhomogeneities in the nebula. We investigated the cause of this discrepancy by constructing photoionization models of increasing complexity. In particular, we used the constraints from the Halpha and Hbeta surface brightness distributions and state-of-the-art models of the stellar ionizing spectrum. None of the successive attempts was able to reproduce the discrepancy between the temperature diagnostics, so the thermal balance of NGC 588 remains unexplained. We give an estimate of the effect of this failure on the O/H and Ne/O estimates and show that O/H is known to within ±0.2 dex.