Carbon nanotubes provide a rare access point into the plasmon physics of one-dimensional electronic systems. By assembling purified nanotubes into uniformly sized arrays, we show that they support coherent plasmon resonances, that these plasmons couple to nanotube and substrate phonons, and that the resulting phonon-plasmon resonances have quality factors as high as 10. Because nanotube plasmons intensely strengthen electromagnetic fields and light-matter interactions, they provide a compelling platform for surface-enhanced spectroscopy and tunable optical devices at deep-subwavelength scales.