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Elastic scattering measurements have been carried out on the ${\mathrm{C}}^{12}$+${\mathrm{C}}^{12}$ and ${\mathrm{O}}^{16}$+${\mathrm{O}}^{16}$ systems in the energy range from 6 to 35 Mev using heavy-ion beams from the Chalk River tandem accelerator and Au-Si surface barrier detectors. At energies below the Coulomb barriers the Mott scattering predictions are in excellent accord with the measurements as functions of both angle and energy. At energies above the barrier the O+O excitation curve drops exponentially below the Mott predictions to a value of 10 mb/sr at ${E}_{\mathrm{c}.\mathrm{m}.}=16.5$ Mev and remains approximately constant thereafter at that value; in contrast, the C+C excitation curve shows marked resonant interference structure. The states involved have $\ensuremath{\tau}\ensuremath{\gtrsim}{10}^{\ensuremath{-}21}$ sec, large compound elastic branching ratios $\frac{{\ensuremath{\Gamma}}_{c}}{\ensuremath{\Gamma}}$, and appear to correspond to resonant absorption of high-order partial waves, hence have high angular momentum. It is suggested that these states reflect a quasi-molecular interaction mechanism which is involved in grazing collisions of these nuclei and which is critically dependent upon the structure of the nuclei involved.