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Carbon nanotubes (CNTs) exhibit highly complicated structures when incorporated in composites and are neither perfectly straight nor aligned in any one direction. Recently, the "clumpy" structure of multiple aggregated CNTs has been likened to desert tumbleweeds. The goal of this paper is to quantify the electromagnetic scattering characteristics of CNTs in the tumbleweed configuration using classical electromagnetic simulations. The simulation results show that the first-plasmon resonance, exhibited by an individual CNT, splits into multiple resonances when multiple CNTs agglomerate into tumbleweed configurations. The number of the resonances increases linearly with the number of CNTs inside the tumbleweed. The second-plasmon resonance exhibits similar splitting behavior. This splitting in the plasmon resonances is caused by the asymmetry of shapes of CNTs and their proximity to each other. If sufficient CNTs agglomerate, the split first-plasmon and second-plasmon resonances overlap, leading to an overall broadband behavior. These results indicate the importance of precise modeling of CNT shapes and the distribution of shapes on the electromagnetic response. Our computations qualitatively explain the broadband response of CNT films and composites and should aid in the development of novel broadband materials and devices based on CNTs.