A projective, normal variety, X , is called a Mori dream space (MDS) when its Cox ring, Cox(X), is finitely generated. While Mori dream spaces have nice behavior, no complete classification of them yet exists. Due to their combinatorial nature, one natural class of candidates for Mori dream spaces is projectivized toric vector bundles. In 2012, Jose Gonzalez proved that all rank-2 projectivized toric vector bundles are MDS. Kaveh and Manon (2019) gave a combinatorial description of toric vector bundles that we use to describe a family of rank-r toric vector bundles that are MDS. This description, along with a relationship with toric full flag bundles, also allows us to describe conditions under which a direct sum of MDS bundles are also MDS. We conclude with computational examples of bundles over products of projective space and directions for future research, including an algorithmic implementation.