We show that the phonon coupling to the electronic system in individual metallic single-walled carbon nanotubes is not due to coupling to low-energy plasmons. The evidence stems from the measured Raman-Stokes G-mode, which for metallic and semiconducting tubes could be fitted well by the superposition of only two Lorentzian lines associated with vibrational modes along the nanotube axis and the nanotube circumference. In the case of metallic tubes the lower-energy G mode is significantly broadened, however maintaining the Lorentzian line shape, in contrast to the theoretically expected asymmetric Breit-Wigner-Fano line shape from phonon-plasmon coupling. The results were obtained by studying 25 individual metallic and semiconducting single-walled carbon nanotubes with atomic force microscopy, electron transport measurements, and resonant Raman spectroscopy.