The series of CH...O bonds formed between CFnH4-n(n = 0-3) and water are studied by quantum calculations under vacuum and in various solvents, including aqueous environment. The results are compared with the OH...O bond of the water dimer in the same solvents. Increasing polarity of the solvent leads in all cases to a lessening of the H-bond interaction energy, in a uniform fashion such that the CH...O bonds all remain weaker than OH...O in any solvent. These H-bond weakenings are coupled to a shortening of the inter-subunit separation. The contraction of the covalent CH bond to the bridging proton is reduced as the solvent becomes more polar, and the blue shift of its stretching vibration is likewise diminished. A process is considered that simulates protein folding by starting from a pair of noninteracting subunits in aqueous solvent and then goes to a H-bonded pair within the confines of a protein environment. This process is found to be energetically more favorable for some of the CH...O H-bonds than for the nominally stronger conventional OH...O H-bond. This finding suggests that CH...O bonds can make important energetic contributions to protein folding, on par with those made by traditional H-bonds.