Some of the most exquisitely precise experimental measurements in all of physics are the masses of the various hadrons.
Consider these examples: the proton mass is known to eleven significant digits. The neutron mass is known to eight. The charged pion mass is known to seven, and so is the neutral pion mass. The neutral D meson mass, the charged D meson mass and the Ds meson mass are known to six. The eta meson mass, the charged kaon mass, the neutral kaon mass, the neutral B meson mass and the charged B meson mass are known to five.
We think we know, in principal, at least, the exact terms of the entire Lagrangian of the Standard Model, and could fit it on a (big) t-shirt. The beta function of each of the Standard Model physical constants can be calculated, in principle, without any experimental input (and usually are calculated that way in practice).
In principle, the hadron masses are largely a function of the quark masses and the strong force coupling constant. The physical constants used to calculate the tweaks to those mass estimates from electromagnetic and weak force effects are also known very precisely (to eight significant digits or more). We can also exploit QCD sum rules, using combinations of existing measurements, including, but not limited to the hadron mass measurements.
But, the up and down quark masses are known to just one significant digit of accuracy, and the strong force coupling constant and the quark masses are known only to three or four significant digits. Also, while these values aren't as precise as we like, we can refine our techniques to reflect the narrow area of parameter space that is allowed by existing measurements of these parameters, in our efforts to determine the values of these physical constants more precisely.
So, why is it that we can't reverse engineer the exactly measured hadron masses, using equations that we know exactly, at least in principle, to get more precise values for the strong force coupling constant and the quark masses? Is this simply a matter of insufficient computational power devoted to the problem? If not, what are the stumbling blocks?