Single-crystal elastic moduli, Cij, and the B1-B2 phase transition of NaCl were investigated experimentally, using time-domain Brillouin scattering (TDBS), and theoretically, via density-functional-theory (DFT), to 41 GPa. Thus, we largely extended pressure range where Cij and elastic anisotropy of the solid are measured, including the first experimental data for the high-pressure B2 phase, NaCl-B2. NaCl-B1 exhibits a strong and growing with pressure anisotropy, in contrast to NaCl-B2. Theoretical values obtained using different advanced DFT functionals were compared with our measurements but no one could satisfactorily reproduce our experimental data for NaCl-B1 and NaCl-B2 simultaneously. For all available DFT results on the principal shear moduli and anisotropy, the deviation became pronounced when the degree of compression increased significantly. Similar deviations could be also recognized for other cubic solids having the same B1-type structure and similar bonding, such as CaO, MgO, or (Mg1-x,Fex)O. Furthermore, the available experimental data suggest that the B1-B2 phase transition of NaCl and the above mentioned compounds are governed by the Born stability criterion C44(P) - P > 0.