alt Hacker NewsLattice-QCD can, by now, actually calculate the masses of the proton, neutron from first principles pretty accurately.
This is of course a brute-force approach. We currently lack, in all fields, theory for emergent properties. And the mass of the proton definitely is such.
There have been claims about this, starting with "Ab Initio Determination of Light Hadron Masses" (Science, 2008).
Nevertheless, until now I have not seen anything that qualifies as "computing the masses".
Research papers like that do not contain any information that would allow someone to verify their claims. Moreover, such papers are much more accurately described as "fitting the parameters of the Standard Model, such as quark masses, to approximately match the measured masses", and not as actually computing the masses.
The published results of hadron masses are not much more accurate than you could compute mentally, without using any QCD, much less Lattice QCD, by estimating approximate quark masses from the composition in quarks of the hadrons and summing them. What complicates the mass computations is that while the heavy quarks have masses that do not vary much, the effective masses of the light quarks (especially u and d, which compose the protons and neutrons) vary a lot between different particles. Because of this, there is a very long way between a vague estimate of the mass and an accurate value.