The support of a \(t\)-design is the set of all distinct blocks in the design. The notation \(t-(v,k, \lambda|b^*)\) is used to denote a \(t\)-design with precisely \(b^*\) distinct blocks. We present some results about the structure of support in \(t\)-designs. Some of them are about the number and the range of occurrences of \(i\)-sets (\(1 \leq i \leq t\)) in the support. A new bound for the support sizes of \(t\)-designs is presented. In particular, given a \(t-(v, k, \lambda|b^*)\) design with \(b > b_0\), where \(b\) and \(b_0\) are the cardinality and the minimum cardinality of block sets in the design, respectively, then it is shown that \(b^* \geq \lceil \frac{\lceil \frac{2b}{\lambda}\rceil +7}{2}\rceil\). We also show that when \(\lambda\) varies over all positive integers, then there is no \(t-(v,k,\lambda | b^*)\)-design with the support sizes equal to \(b^*_{min}+1, b^*_{min}+2\) and \(b^*_{min}+3\), where \(b^*_{min}\) denotes the least possible cardinality of the support sizes in this design.