A combined experimental and theoretical spectroscopic study of high-n, 30≲n≲100, triplet S and D Rydberg states in 87Sr is presented. 87Sr 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 S states, the hyperfine shifts are evaluated nonperturbatively, taking advantage of earlier spectroscopic data for the I=0 isotope 88Sr, which results in good agreement with the present measurements. For the D states, this procedure is reversed by first extracting from the present 87Sr measurements the energies of the 3D1,2,3 states to be expected for isotopes without hyperfine structure (88Sr), which allows the determination of corrected quantum defects in the high-n limit.