The excited states of polycyclic aromatic hydrocarbons (PAH) with up to 28 π-electrons have been investigated in the framework of time-dependent density functional theory (TDDFT). Calculations of the two lowest-lying singlet-singlet and the lowest-lying singlet-triplet excitation energies for sixteen molecules of very different structure are used to assess the accuracy and applicability of the method. The gradient corrected BP86 and hybrid-type B3LYP functionals together with the large cc-pVTZ AO basis set have been used. In general it is found that TDDFT should be used with caution for the singlet states of these large unsaturated π-systems. The TDDFT approach underestimates the excitation energies for states with dominant ionic character ( 1 L a ) with a mean absolute deviation (MAD) of 0.18 eV (B3LYP) and 0.49 eV (BP86), respectively. The MAD for covalent 1 L b (B3LYP: 0.24 eV, BP86: 0.12 eV) and 3 L a (B3LYP: 0.04 eV, BP86: 0.05 eV) states are significantly smaller. The TDDFT results are also compared with those from a semi-empirical time-dependent Pariser-Parr-Pople (TDPPP) treatment which is found to be very accurate (L a / b MAD<0.16 eV) for the systems considered. The TDDFT errors are analyzed in a simplified valence-bond picture derived from the TDPPP data. It is shown that the TDDFT errors correlate linearly with the estimated ioniticity of the states which may open a route to correct TDDFT results or to improve existing functionals.