  We investigate terminal-pairability properties of complete graphs and improve
the known bounds in two open problems. We prove that the complete graph $K_n$
on $n$ vertices is terminal-pairable if the maximum degree $\Delta$ of the
corresponding demand graph $D$ is at most $2\lfloor\frac{n}{6}\rfloor-2$. We
also verify the terminal-pairability property when the number of edges in $D$
does not exceed $2n-5$ and $\Delta\leq n-1$ holds.
