We prove that the set of-y-thick points of a planar Gaussian free field (GFF) with Dirichlet boundary conditions is a.s. totally disconnected for all-y =6 0. Our proof relies on the coupling between a GFF and the nested CLE4. In particular, we show that the thick points of the GFF are the same as those of the weighted CLE4 nesting field introduced in [24] and establish the almost sure total disconnectedness of the complement of a nested CLE & kappa;, & kappa; & ISIN; (8/3, 4]. As a corollary we see that the set of singular points for supercritical LQG metrics is a.s. totally disconnected.