Let $\lambda, \mu$ be weights of $G = SL(n,\mathbb{C})$ such that $\lambda$ is dominant. Let $V_\lambda[\mu]$ denote the $\mu$--isotypic component of the irreducible representation $V_\lambda$ with highest weight $\lambda$. Let $R(\lambda,\mu) = \bigoplus_{N=0}^\infty V_{N \lambda}[N \mu]$. Then $R(\lambda,\mu)$ has a natural ring structure, graded by $N$; it is the coordinate ring of the \emph{weight variety} $W(\lambda,\mu) = \text{Proj}(R(\lambda,\mu))$, which is a geometric invariant theory quotient of the space of full flags in $\mathbb{C}^n$ by the action of a maximal torus in $G$. Assuming $V_\lambda[\mu]$ is nonzero, we show that $R(\lambda,\mu)$ is generated in degree $\leq \dim(W(\lambda,\mu))$. These rings include rings of projective invariants of configurations of weighted points in projective space. By work of Lakshmibai-Gonciulea, there is a flat degeneration of $R(\lambda,\mu)$ to the coordinate ring $R'(\lambda,\mu)$ of a toric variety. The ring $R'(\lambda,\mu)$ is the semigroup algebra of a certain set of Gelfand Tsetlin patterns (the semigroup operation is addition of patterns). We show that $R'(\lambda,\mu)$ can require generators of degree exponential in $n$; in particular this is true for point configurations in $\mathbb{CP}^2$.