Context. Diffuse interstellar absorption bands (DIBs) of largely unknown chemical origin are regularly observed primarily in distant early-type stars. More recently, detections in nearby late-type stars have also been claimed. These stars' spectra are dominated by stellar absorption lines. Specifically, strong interstellar atomic and DIB absorption has been reported in τ Boo. Aims. We test these claims by studying the strength of interstellar absorption in high-resolution TIGRE spectra of the nearby stars τ Boo, HD 33608, and α CrB. Methods. We focus our analysis on a strong DIB located at 5780.61 Å and on the absorption of interstellar Na. First, we carry out a differential analysis by comparing the spectra of the highly similar F-stars, τ Boo and HD 33608, whose light, however, samples different lines of sight. To obtain absolute values for the DIB absorption, we compare the observed spectra of τ Boo, HD 33608, and α CrB to PHOENIX models and carry out basic spectral modeling based on Voigt line profiles. Results. The intercomparison between τ Boo and HD 33608 reveals that the difference in the line depth is 6.85 ± 1.48 mÅ at the DIB location which is, however, unlikely to be caused by DIB absorption. The comparison between PHOENIX models and observed spectra yields an upper limit of 34.0 ± 0.3 mÅ for any additional interstellar absorption in τ Boo; similar results are obtained for HD 33608 and α CrB. For all objects we derive unrealistically large values for the radial velocity of any presumed interstellar clouds. In τ Boo we find Na D absorption with an equivalent width of 0.65 ± 0.07 mÅ and 2.3 ± 0.1 mÅ in the D 2 and D 1 lines. For the other Na, absorption of the same magnitude could only be detected in the D 2 line. Our comparisons between model and data show that the interstellar absorption toward τ Boo is not abnormally high. Conclusions. We find no significant DIB absorption in any of our target stars. Any differences between modeled and observed spectra are instead attributable to inaccuracies in the stellar atmospheric modeling than to DIB absorption.