Aligned densely packed carbon-nanotube metamaterials prepared using vacuum filtration are an emerging infrared nanophotonic material. We report multiple hyperbolic plasmon resonances, together spanning the mid-infrared, in individual resonators made from aligned and densely packed carbon nanotubes. In a near-field scanning optical microscopy (NSOM) imaging study of nanotube metamaterial resonators, we observe distinct deeply subwavelength field profiles at the fundamental and higher-order resonant frequencies. The wafer-scale area of the nanotube metamaterials allows us to combine this near-field imaging with a systematic far-field spectroscopic study of the scaling properties of many resonator arrays. Thorough theoretical modeling agrees with these measurements and identifies the resonances as higher-order Fabry-Perot (FP) resonances of hyperbolic waveguide modes. Nanotube resonator arrays show broadband extinction from 1.5-10 μm and reversibly switchable extinction in the 3-5 μm atmospheric transparency window through the coexistence of multiple modes in individual ribbons. Broadband carbon-nanotube metamaterials supporting multiple resonant modes are a promising candidate for ultracompact absorbers, tunable thermal emitters, and broadband sensors in the mid-infrared.