Low orbital overlap and triplet instability problems in time-dependent density functional theory (TDDFT) are investigated for a new benchmark set, encompassing challenging singlet and triplet excitation energies of local, charge-transfer, and Rydberg character. The low orbital overlap problem is largely overcome for both singlet and triplet states by the use of a Coulomb-attenuated functional. For all the categories of functional considered, however, errors associated with triplet instability problems plague high overlap excitations, as exemplified by the excited states of acenes and polyacetylene oligomers. Application of the Tamm-Dancoff approximation reduces these errors for both singlet and triplet states, while leaving low-overlap excitations unaffected. The study illustrates the synergy between overlap and stability and highlights the success of a combined, Coulomb-attenuated Tamm-Dancoff approach.