We investigate the mixing time of the capacity $k$ simple exclusion process (also called the partial exclusion process) of Schultz and Sandow with $m$ particles on a segment of length $N$. We show that the $k$-SEP exhibits cutoff at time $\frac{1}{2kπ^2}N^2\log m$. We also introduce a related complete multi-species process that we call the $S_{k,N}$ shuffle and show that this process exhibits cutoff at time $\frac{1}{2kπ^2}N^2\log (kN)$. This extends the celebrated result of Lacoin which determined the mixing time of the symmetric simple exclusion process on a segment of length $N$ and the adjacent transposition shuffle, and proved cutoff in both.