The tunneling density of states has been studied on disordered multiwalled carbon nanotubes. The tunneling conductance shows a large zero-bias anomaly, whose temperature and voltage dependence is successfully compared with the non-perturbative theory of electron tunneling into a disordered ID electrode. The environmental Coulomb blockade is expected to set in at lower energies, where junctions can be considered to be zero-dimensional. In one of the samples, Coulomb blockade behavior is revealed over a wide range of temperatures. In this sample the tunneling is also studied using a superconducting counterelectrode, and the observed reduction of the conductivity is found to be in quantitative agreement with the theory.