Let \(E\) be a locally convex space. The diametral dimension of \(E\) is the set \[ \Delta(E)=\{\xi\in\mathbb C^{\mathbb N_0}: \forall V\in \mathcal{V}(E)\;\exists U\in \mathcal{V}(E)\;\text{such\;that}\;U\subset V \;\text{and }\;(\xi_n \delta_n(U,V))_{n\in\mathbb N_0}\in c_0 \}, \] where \(\mathcal{V}(E)\) is a basis of \(0\)-neighbourhoods in \(E\) and \(\delta_n(U,V)\) is the \(n\)th Kolmogorov diameter of \(U\) with respect to \(V\), i.e., \[ \delta_n(U,V)=\inf\{\delta>0: \exists F\in \mathcal{L}_n(E) \;\text{such\;that }\;U\subset \delta V+F\}, \] where \(\mathcal{L}_n(E)\) denotes the set of all linear subspaces of \(E\) with a dimension less or equal to \(n\). \textit{T. Terzioglu} [Collect. Math. 20, 49--99 (1969; Zbl 0175.41602)] introduced another definition of diametral dimension as follows: \[ \Delta_b(E)=\{\xi\in\mathbb C^{\mathbb N_0}: \forall V\in \mathcal{V}(E)\;\forall B\in \mathcal{B}(E)\;\text{such\;that}\;(\xi_n \delta_n(B,V))_{n\in\mathbb N_0}\in c_0 \}, \] where \(\mathcal{B}(E)\) denotes the set of all bounded subsets of \(E\). Obviously, \(\Delta(E)\subseteq\Delta_b(E)\) and \(\Delta(E)=\Delta_b(E)\) in the case that \(E\) is a normed space. But it is known that, if \(E\) is a Fréchet-Montel space which is not Schwartz, then \(\Delta(E)=c_0\subsetneq\Delta_b(E)\). The question of the equality \(\Delta(E)=\Delta_b(E)\) in the case of non-normable locally convex spaces remains an open problem even in the case of Fréchet-Schwartz spaces \(E\). In the paper under review, the authors give sufficient conditions for the equality \(\Delta(E)=\Delta_b(E)\). They apply these conditions to Köthe echelon spaces defined with a regular Köthe matrix, to Köthe echelon spaces with property \((\overline{\Omega})\), and to Köthe echelon spaces of type \(G_\infty\). Moreover, the authors construct examples of nuclear (hence Schwartz) non-metrizable locally convex spaces for which the equality fails.