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An operator $T$ on a Banach space $X$ is ${\it hypercyclic}$ (respectively, ${\it supercyclic}$) if there exists a vector $x$ in $X$ for which its orbit Orb$(T,x) = \{T^n x : n \ge 0 \}$ (respectively, the projective orbit $\mathbb{C}\cdot$Orb$(T,x) = \{\lambda T^n x : n \ge 0 \mbox{ and } \lambda \in \mathbb{C}\}$) is dense in $X$. In 2007, Bes and Peris, and Bernal-Gonzalez independently introduced the concept of disjoint hypercyclicity. We say two or more linear operators $T_1, \ldots, T_N$ are ${\it disjoint\; hypercyclic}$ (respectively, ${\it disjoint\; supercyclic}$) if there exists a vector $x$ in $X$ for which the direct sum $T_1 \oplus \dots \oplus T_N$ has a hypercyclic (respectively, supercyclic) vector in the form $(x, \ldots, x) \in X^N$. In the class of weighted shift operators, we partially answer a question of Salas asking if given a finite collection $T_1, \ldots, T_N$ of disjoint hypercyclic (supercyclic) operators, can one find an additional operator $T_{N+1}$ for which the larger family $T_1, \ldots, T_N, T_{N+1}$ remains disjoint hypercyclic (supercyclic). To this end, we first characterize disjoint hypercyclic and supercyclic weighted shifts in terms of their weight sequences.