A sequence of $S_n$-representations $\{V_n\}$ is said to be uniformly representation stable if the decomposition of $V_n = \bigoplus_μ c_{μ,n} V(μ)_n$ into irreducible representations is independent of $n$ for each $μ$---that is, the multiplicities $c_{μ,n}$ are eventually independent of $n$ for each $μ$. Church-Ellenberg-Farb proved that the cohomology of flag varieties (the so-called diagonal coinvariant algebra) is uniformly representation stable. We generalize their result from flag varieties to all Springer fibers. More precisely, we show that for any increasing subsequence of Young diagrams, the corresponding sequence of Springer representations form a graded co-FI-module of finite type (in the sense of Church-Ellenberg-Farb). We also explore some combinatorial consequences of this stability.