We investigate the accuracy of different formalisms within density functional theory in prediction of two-photon absorption (2PA) spectra for substituted oligophenylvinylenes compared to the experimental measurements. The quadratic response methods are compared with the recently proposed a posteriori Tamm-Dancoff approximation (ATDA) and previously published third-order coupled electronic oscillator results. Quadratic response is found to overestimate the cross sections in all cases. We trace the reasons to unreliable excited state description above the ionization threshold. In addition, quadratic response lacks the double excitations so that their contributions to the 2PA spectra are redistributed over the nearest single character excitations. This distorts the individual contributions to the 2PA response and affects the overall picture. For this reason, we do not recommend quadratic response for the essential state analysis, while ATDA can be used both for the 2PA predictions and the structure/property correlations. As an illustration for ATDA based essential state analysis, we report the mechanism of large 2PA in symmetric donor/acceptor substituted polyphenylvinylene (PPV) oligomers. While HOMO-LUMO transition provides the only bright intermediate state, the brightness of the one-photon absorption (1PA) to 2PA transition is associated with symmetric to asymmetric linear combination of the respective donor (HOMO -1 to HOMO) or acceptor (LUMO to LUMO + 1) fragment orbitals of the donor or acceptor substituents. We also study the effect of the fraction of Hartree-Fock (HF) exchange on 2PA excitation energies and cross sections. Higher exchange (BMK and M05-2X) and range separated (CAM-B3LYP) hybrid functionals are found to yield rather inaccurate predictions both quantitatively and qualitatively. The results obtained with the long-range corrected functional LC-BLYP do not seem to be useful at all. This failure of the exchange-correlation functionals with the correct asymptotic is traced to inaccurate transition dipoles between the valence states, where only functionals with lower HF exchange succeed. A new sum over states (SOS) cutoff procedure is proposed to compensate for the collapse of the higher-lying excited states obtained with the hybrid functionals.