There are two ways of turning Segal spaces into models for up-to-homotopy categories, or $(\infty,1)$-categories: either asking that the space of objects be discrete, or requiring Rezk's completeness condition. When generalizing to higher $(\infty,n)$-categories, both of these approaches have been taken to multisimplicial models, in the form of Segal $n$-categories and $n$-fold complete Segal spaces, but models given by $\Theta_n$-diagrams have focused on the completeness conditions. In this talk, we'll discuss how to get a $\Theta_n$-model with discreteness conditions, but also address the question of when these conditions can be mixed and matched with one another.