Quantum chemistry has reached a point where some gas phase properties of small molecules can be computed more accurately than they can be measured. For these systems science can say "problem solved", give itself a light pad on the back, and move on to other things - like how to do the same for molecules twice the size. Or in solution.
I don't believe there is an equivalent success story in biomolecular modeling, but, as I'll argue here, the prediction of the pKa values of ionizable residues in proteins be a close contender!
Jens Erik Nielsen and co-worker recently published a very interesting paper entitled "Remeasuring HEWL pKa values by NMR spectroscopy: Methods, analysis, accuracy, and implications for theoretical pKa calculations" in Proteins.
One of the most interesting implications of the paper, in my opinion, can be found at the bottom of Table III. Here the authors compare Asp, Glu, and His pKa values in HEWL measured using the NMR chemical shifts of backbone nuclei to those obtained from side chain 13C atoms.
The RMSD values are 1.3, 1.2. and 0.6 pH units for 15N, 1HN, and 1Hα, respectively (note that the latter could only be obtained for 7 of the 11 residues). Now compare this to the corresponding values for pKa values predicted with the pKD and PROPKA servers: 0.9 and 0.8.
Thus, if you are not willing to go through the considerable trouble that is assigning side chain chemical shifts for a protein, you are better off predicting the pKa values than measuring them! At least for this protein ... and only if you can't live without knowing all the pKa values ... but still.
Nielsen and co-workers estimate the accuracy of the 13C-derived to be accurate to within 0.1 - 0.2 pH units, so, no, the general problem is not solved. But if more people do as careful a job measuring pKa values as Jens, we may find that we are closer than we think.