A combined experimental and theoretical spectroscopic study of high-$n$, ${30 lesssim n lesssim 100}$, triplet $text{S}$ and $text{D}$ Rydberg states in $^{87}text{Sr}$ is presented. $^{87}text{Sr}$ has a large nuclear spin, ${I=9/2}$, and at high-$n$ the hyperfine interaction becomes comparable to, or even larger than, the fine structure and singlet-triplet splittings which poses a considerable challenge both for precision spectroscopy and for theory. For high-$n$ $text{S}$ states, the hyperfine shifts are evaluated non-perturbatively taking advantage of earlier spectroscopic data for the ${I=0}$ isotope $^{88}text{Sr}$, which results in good agreement with the present measurements. For the $text{D}$ states, this procedure is reversed by first extracting from the present $^{87}text{Sr}$ measurements the energies of the $^{3}text{D}_{1,2,3}$ states to be expected for isotopes without hyperfine structure ($^{88}text{Sr}$) which allows the determination of corrected quantum defects in the high-$n$ limit.
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