We demonstrate that the atomic alignment of the hyperfine-structure components of the ground-level S1/2 of Na I and of the upper-level P1/2 of the D1 line are practically negligible for magnetic strengths B>10 G and virtually zero for B>~100 G. This occurs independently of the magnetic field inclination on the stellar surface (also, in particular, for vertical fields). Consequently, the characteristic antisymmetric linear polarization signature of the scattered light in the D1 line is practically suppressed in the presence of magnetic fields larger than 10 G, regardless of their inclination. Remarkably, we find that the scattering polarization amplitude of the D2 line increases steadily with the magnetic strength, for vertical fields above 10 G, while the contribution of the alignment to the polarization of the D1 line rapidly decreases. Therefore, we suggest that spectropolarimetric observations of the ``quiet'' solar chromosphere showing significant linear polarization peaks in both D1 and D2 cannot be interpreted in terms of one-component magnetic field models, implying that the magnetic structuring of the solar chromosphere could be substantially more complex than previously thought.