We report a new search for 12CO(1-0) emission in high-velocity clouds (HVCs) performed with the IRAM 30 m telescope. This search was motivated by the recent detection of cold dust emission in the HVCs of Complex C. Despite a spatial resolution which is three times better and sensitivity twice as good compared to previous studies, no CO emission is detected in the HVCs of Complex C down to a best 5 sigma limit of 0.16 K km/s at a 22 resolution. The CO emission non-detection does not provide any evidence in favor of large amounts of molecular gas in these HVCs and hence in favor of the infrared findings. We discuss different configurations which, however, allow us to reconcile the negative CO result with the presence of molecular gas and cold dust emission. H2 column densities higher than our detection limit, N(H2) = 3x10^{19} cm^{-2}, are expected to be confined in very small and dense clumps with 20 times smaller sizes than the 0.5 pc clumps resolved in our observations according to the results obtained in cirrus clouds, and might thus still be highly diluted. As a consequence, the inter-clump gas at the 1 pc scale has a volume density lower than 20 cm^{-3} and already appears as too diffuse to excite the CO molecules. The observed physical conditions in the HVCs of Complex C also play an important role against CO emission detection. It has been shown that the CO-to-H2 conversion factor in low metallicity media is 60 times higher than at the solar metallicity, leading for a given H2 column density to a 60 times weaker integrated CO intensity. And the very low dust temperature estimated in these HVCs implies the possible presence of gas cold enough (< 20 K) to cause CO condensation onto dust grains under interstellar medium pressure conditions and thus CO depletion in gas-phase observations.
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