To address the updated question, a family of subsets of $[n]$ is called *$k$-Sperner* if it does not contain a chain of length $k+1$.  By taking all sets whose size lies in the middle $k$ values of $[n]$, there exist $k$-Sperner families who size is the sum of the $k$ middle bịnomial coefficients. Erdős proved that this bound is tight in this [paper](https://projecteuclid.org/euclid.bams/1183507531)  (see Theorem 5).  The extremal example is also essentially unique (for parity reasons there may be two intervals of middle $k$ values).