We consider the impact of varying the aspect ratio A of carbon nanotubes (CNT) on the theological properties of isotactic polypropylene/CNT nanocomposites. Specifically, we focus on multiwall CNT having a relatively low aspect ratio (A in the range from 22 to 45), since previous work (Kharchenko et al., Nature Mater. 2004) has emphasized the rheological properties of relatively high aspect ratio multiwall CNT nanocomposites (A from 300 to 400). Correspondingly, we find that the formation of a nonequilibrium CNT network structure occurs at a higher CNT concentration in our low A CNT nanocomposites, as theoretically expected. The proposed mechanism for the large apparent negative normal stress differences (Delta N) described by Kharchenko et al. relies on the capacity of the "struts" of the CNT network to rotate about their impingement junctions, much like the links of a deformed chain link fence. This model implies the absence of appreciable negative Delta N at low CNT concentrations where the network does not yet exist and also for short CNT and at high concentrations of CNT where the mesh size of the network becomes too small to accommodate appreciable rotational distortion. In conformity with this simple mechanical model, we observe only a positive apparent Delta N in our CNT/iPP` nanocomposites, even well above the CNT gelation concentration. This striking change in the rheology of CNT nanocomposites with a change in A has been further confirmed in die-swell measurements, where a large die swell has been seen in the short CNT nanocomposites, rather than the die-shrinkage found before for the large A or the highly "entangled" CNT network counterpart.