We report experimental ¹²⁵Te magic-angle spinning solid-state nuclear magnetic resonance (MAS NMR) measurements of the tellurium chemical shift (CS) tensors in three [K(18-crown-6)]⁺ 3,4-dicyano-1,2,5-telluradiazole-XCN⁻ (X = O, S, Se) salt cocrystals featuring chalcogen bonds. These data are compared to those for pure 3,4-dicyano-1,2,5-telluradiazole (1). A reduction in the span of the ¹²⁵Te CS tensor is consistently noted in the salt cocrystals compared to pure 1. Isotopically ¹⁵N-labelled [K(18-crown-6)]⁺[1-OC¹⁵N]⁻, which features a chalcogen bond between Te and the cyanate nitrogen atom, is synthesized using KOC¹⁵N, and the nitrogen CS tensors are measured for both samples via ¹⁵N slow MAS NMR spectroscopy. Possible dynamic disorder of the cyanate ions in KOCN is ruled out. Two crystallographically distinct nitrogen sites are resolved for the salt cocrystal. Upon formation of [K(18-crown-6)]⁺[1-OC¹⁵N]⁻, the ¹⁵N isotropic CS and CS tensor span both decrease relative to the values for pure KOC¹⁵N, and the axial symmetry of this tensor is lost. These findings are supplemented with a series of density functional theory calculations of magnetic shielding tensors using cluster models or periodic boundary conditions. Inclusion of spin–orbit relativistic effects in the calculation of tellurium shielding tensors is particularly important in achieving agreement with experiment.