We investigate the broadband dielectric properties of vertically aligned, multi-wall carbon nanotubes (VACNT), over both the terahertz (THz) and mid-infrared spectral ranges. The nominally undoped, metallic VACNT samples are probed at normal incidence, i.e. the response is predominantly due to polarisation perpendicular to the CNT axis. A detailed comparison of various conductivity models and previously reported results is presented for the non-Drude behaviour we observe in the conventional THz range (up to 2.5 THz). Extension to the mid-infrared range reveals an absorption peak at ∼24 THz, reminiscent of that observed in single-wall CNT, only there it arises for polarisation parallel to the CNT axis. To account for the observed resonance here, we apply a Bergman-type effective-medium theory, based on first-principles' electromagnetic simulations for the perpendicular polarisation including both the intra- and inter-tube response, which can reproduce the observed spectrum if one assumes a much higher plasma frequency and scattering rate than that reflected in the low-frequency spectra, and proposes an explanation for the non-Drude behaviour at low-frequencies. © 2017 IOP Publishing Ltd.