  Given two $n$-vertex graphs $G_1$ and $G_2$ of bounded treewidth, is there an
$n$-vertex graph $G$ of bounded treewidth having subgraphs isomorphic to $G_1$
and $G_2$? Our main result is a negative answer to this question, in a strong
sense: we show that the answer is no even if $G_1$ is a binary tree and $G_2$
is a ternary tree. We also provide an extensive study of cases where such
`gluing' is possible. In particular, we prove that if $G_1$ has treewidth $k$
and $G_2$ has pathwidth $\ell$, then there is an $n$-vertex graph of treewidth
at most $k + 3 \ell + 1$ containing both $G_1$ and $G_2$ as subgraphs.
