We show that the electronic circular dichroism (ECD) of delocalized π-systems represents a worst-case scenario for Tamm-Dancoff approximated (TDA) linear response methods. We mainly consider density functional theory (TDA-DFT) variants together with range-separated hybrids, but the conclusions also apply for other functionals as well as the configuration interaction singles (CIS) approaches. We study the effect of the TDA for the computation of ECD spectra in some prototypical extended π-systems. The C76 fullerene, a chiral carbon nanotube fragment, and [11]helicene serve as model systems for inherently chiral, π-chromophores. Solving the full linear response problem is inevitable in order to obtain accurate ECD spectra for these systems. For the C76 fullerene and the nanotube fragment, TDA and CIS approximated methods yield spectra in the origin-independent velocity gauge formalism of incorrect sign which would lead to the assignment of the opposite (wrong) absolute configuration. As a counterexample, we study the ECD of an α-helix polypeptide chain. Here, the lowest-energy transitions are dominated by localized excitations within the individual peptide units, and TDA methods perform satisfactorily. The results may have far-reaching implications for simple semiempirical methods which often employ TDA and CIS for huge molecules. Our recently presented simplified time-dependent DFT approach proves to be an excellent low-cost linear response method which together with range-separated density functionals like ωB97X-D3 produces ECD spectra in very good agreement with experiment.