We demonstrate that the atomic alignment of the hyperfine-structure components of the ground level S$_{1/2}$ of Na {\sc i} and of the upper level P$_{1/2}$ of the D$_1$ line are practically negligible for magnetic strengths $B>10 \rm G$, and virtually zero for $B\ga 100 \rm G$. 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 D$_1$ 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 D$_2$ line increases steadily with the magnetic strength, for vertical fields above 10 G, while the contribution of alignment to the polarization of the D$_1$ line rapidly decreases. Therefore, we suggest that spectropolarimetric observations of the ``quiet'' solar chromosphere showing significant linear polarization peaks in both D$_1$ and D$_2$ 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. Comment: 11 pages and 2 figures. The Astrophysical Journal Letter (in press)