In this paper, we consider classes of decision tables closed under removal of attributes (columns) and changing of decisions attached to rows. For decision tables from closed classes, we study lower bounds on the minimum cardinality of reducts, which are minimal sets of attributes that allow us to recognize, for a given row, the decision attached to it. We assume that the number of rows in decision tables from the closed class is not bounded from above by a constant. We divide the set of such closed classes into two families. In one family, only standard lower bounds $Ω(\log $ ${\rm cl}(T))$ on the minimum cardinality of reducts for decision tables hold, where ${\rm cl}(T)$ is the number of decision classes in the table $T$. In another family, these bounds can be essentially tightened up to $Ω({\rm cl}(T)^{1/q})$ for some natural $q$.