Engineered core-shell cylinders are good candidates for applications in invisibility and cloaking. In particular, hyperbolic nanotubes demonstrate tunable ultra-low scattering cross section in the visible spectral range. In this work we investigate the limits of validity of the condition for invisibility, which was shown to rely on reaching an epsilon near zero in one of the components of the effective permittivity tensor of the anisotropic metamaterial cavity. For incident light polarized perpendicularly to the scatterer axis, critical deviations are found in low-birefringent arrangements and also with high-index cores. We demonstrate that the ability of anisotropic metallodielectric nanocavities to dramatically reduce the scattered light is associated with a multiple Fanoresonance phenomenon. We extensively explore such resonant effect to identify tunable windows of invisibility.