We address the conundrum posed by the well-known failure of time-dependent DFT (TDDFT) with conventional functionals for "charge-transfer-like" excitations in oligoacenes. We show that this failure is due to a small spatial overlap in orbitals obtained from the underlying single-electron orbitals by means of a unitary transformation. We further show that, as in true charge-transfer excitations, this necessarily results in failure of linear-response TDDFT with standard functionals. Range-separated hybrid functionals have been previously shown to mitigate such errors but at the cost of an empirically adjusted range-separation parameter. Here, we explain why this approach should succeed where conventional functionals fail. Furthermore, we show that optimal tuning of a range-separated hybrid functional, so as to enforce the DFT version of Koopmans' theorem, restores the predictive power of TDDFT even for such difficult cases, without any external reference data and without any adjustable parameters. We demonstrate the success of this approach on the oligoacene series and on related hydrocarbons. This resolves a long-standing question in TDDFT and extends the scope of molecules and systems to which TDDFT can be applied in a predictive manner.