In this paper, we show that for every sufficiently large integer \(n\) and every positive integer \(c \leq \left\lfloor \frac{1}{6}({\log \log n})^\frac{1}{2} \right \rfloor\), a Boolean lattice with \(n\) atoms can be partitioned into chains of cardinality \(c\), except for at most \(c-1\) elements which also form a chain.