How accurate are the most accurate calculations?

Exhibit 1: Ground state hyperfine splitting of the H atom:

1420405751767(1) mHz (present most accurate experiment)
142045199        mHz (present most accurate theory)

The error in the theory is due to the difficulty in treating the nuclear structure (2 up quarks + 1 down).

Exhibit 2: Ground state hyperfine splitting of the muonium atom:

4463302780(050) Hz (present most accurate experiment)
4463302880(550) Hz (present most accurate theory)

Why does it agree so well? μ$^+$ is a fundamental particle and therefore has no nuclear structure. QED is the correct theory for describing the interaction between pure electric charges (e$^-$ and μ$^+$). The only QFD (quantum flavordynamics) needed is for the electro-weak interaction between the particles (not for interactions within sub-nuclear particles), and QFD calculations were done here in anticipation of more accurate experiments to come.

Exhibit 3: Ground state hyperfine splitting of the He atom:

6739701177(0016) Hz (present most accurate experiment)
6739699930(1700) Hz (present most accurate theory)

Notice how much harder it is when you add an electron.

Exhibit 5: $Srightarrow P$ transition in the Li atom:

14903.632061014(5003) cm^-1 (present most accurate experiment)
14903.631765(0006670) cm^-1 (present most accurate theory)

Exhibit 6: Ionization energy of the Li atom:

43487.15940(18) cm^-1 (present most accurate experiment)
43487.1590(080) cm^-1 (present most accurate theory)

Exhibit 7: Ionization energy of the Be atom:

76192.64(0060) cm^-1 (present most accurate experiment)
76192.699(007) cm^-1 (present most accurate theory)

Notice that theory is 1 order of magnitude more accurate than experiment!!!

Exhibit 8: Atomization energy of the H$_2$ molecule:

35999.582834(11) cm^-1 (present most accurate experiment)
35999.582820(26) cm^-1 (present most accurate theory)

See here for more info.

Exhibit 9: Fundamental vibration of the H$_2$ molecule:

4161.16632(18) cm^-1 (present most accurate experiment)
4161.16612(90) cm^-1 (present most accurate theory)

See here for HD and D$_2$.