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We explore the maximum possible toughness among graphs with \(n\) vertices and \(m\) edges in the cases in which \(\lceil \frac{3n}{2}\rceil \leq m < 2n\). In these cases, it is shown that the maximum toughness lies in the interval \([\frac{4}{3}, \frac{3}{2}]\). Moreover, if \(\left\lceil\frac{3n}{2}\right\rceil + 2 \leq m < 2n\), then the value \(\frac{3}{2}\) is achieved. However, if \(m \in \left\{\left\lceil\frac{3n}{2}\right\rceil, \left\lceil\frac{3n}{2}\right\rceil + 1\right\}\), then the maximum toughness can be strictly less than \(\frac{3}{2}\). This provides an infinite family of graphs for which the maximum toughness is not half of the maximum connectivity. The values of maximum toughness are computed for all \(1 \leq n \leq 12\), and some open problems are presented.