Tree decompositions of optimal width where every vertex is in a bounded number of bags?
Answering my own question: the answer is "no". For each $k \ge 3$, Ding and Oporowski construct a graph $G_k$ on $2k$ vertices with treewidth $3$, such that in every optimal tree decomposition of $G_k$ there is a vertex which is in at least $2k-2$ bags (see Figure 1 in their paper).

Ding, Guoli; Oporowski, Bogdan, Some results on tree decomposition of graphs, J. Graph Theory 20, No. 4, 481-499 (1995). ZBL0837.05044.