A family of subsets of an $n$-set is $2$-cancellative if, for every four-tuple $\{A, B, C, D\}$ of its members, $A \cup B \cup C=A \cup B \cup D$ implies $C = D$. This generalizes the concept of cancellative set families, defined by the property that $A \cup B \neq A \cup C$ for $A, B, C$ all different. The asymptotics of the maximum size of cancellative families of subsets of an n-set is known (Tolhuizen [7]). We provide a new upper bound on the size of $2$-cancellative families, improving the previous bound of $2^{0.458n}$ to $2^{0.42n}$.