DNA nanostructures have emerged as promising nanomedical tools due to their biocompatibility and tunable behavior. Recent work has shown that DNA nanocages decorated with organic dendrimers strongly bind human serum albumin (HSA), yet the dynamic structures of these complexes remain uncharacterized. This theoretical and computational investigation elucidates the fuzzy interactions between dendritically functionalized cubic DNA nanocages and HSA. The dendrimer-HSA interactions occur via nonspecific binding with the protein thermodynamically and kinetically free to cross the open faces of the cubic scaffold. However, the rigidity of the DNA scaffold prevents the binding energetics from scaling with the number of dendrimers. These discoveries not only provide a useful framework by which to model general interactions of DNA nanostructures complexed with serum proteins but also give valuable molecular insight into the design of next-generation DNA nanomedicines.