Extreme-ultraviolet (EUV) spectra of Sn 13+ −Sn 15+ ions have been measured in an electron-beam ion trap (EBIT). A matrix inversion method is employed to unravel convoluted spectra from a mixture of charge states typically present in an EBIT. The method is benchmarked against the spectral features of resonance transitions in Sn 13+ and Sn 14+ ions. Three new EUV lines in Sn 14+ confirm its previously established level structure. This ion is relevant for EUV nanolithography plasma but no detailed experimental data currently exist. We used the Cowan code for first line identifications and assignments in Sn 15+. The collisional-radiative modeling capabilities of the Flexible Atomic Code were used to include line intensities in the identification process. Using the 20 lines identified, we have established 17 level energies of the 4p 4 4d configuration as well as the fine-structure splitting of the 4p 5 ground-state configuration. Moreover, we provide state-of-the-art ab initio level structure calculations of Sn 15+ using the configuration-interaction many-body perturbation code AMBiT. We find that the here-dominant emission features from the Sn 15+ ion lie in the narrow 2% bandwidth around 13.5 nm that is relevant for plasma light sources for state-of-the-art nanolithography.