Structural disorder within self-assembled molecular aggregates may have strong effects on their optical functionality. Such disorder, however, is hard to explore using standard ensemble measurements. In this paper, we report on the characterization of intra-aggregate structural disorder through a linewidth analysis of fluorescence excitation experiments on individual zinc-chlorin (ZnChl) nanotubular molecular aggregates. Recent experiments suggest an anomaly in the linewidths of the two absorption bands that dominate the spectra: the higher-energy bands on average show a smaller linewidth than the lower-energy bands. This anomaly is explored in this paper by analyzing and modeling the correlation of the two linewidths for each aggregate. We exploit a Frenkel exciton model to show that the experimentally observed correlation of linewidths and other statistical properties of the single-aggregate spectra can be explained from small variations of the molecular orientations within individual aggregates.