We report on the giant magnetoresistance (GMR) of multiwall carbon nanotubes with ultrasmall diameters. In particular, we consider the effect of the interwall interactions and the lead/nanotube coupling. Comparative studies have been performed to show that in the case when all walls are well coupled to the electrodes, the so-called inverse GMR can appear. The tendency towards a negative GMR depends on the interwall interaction and on the nanotube length. If, however, the inner nanotubes are out of contact with one of the electrodes, the GMR remains positive even for relatively strong interwall interactions regardless of the outer nanotube length. These results shed additional light on recently reported experimental data, where an inverse GMR was found in some multiwall carbon nanotube samples.