Can a principal ideal contain a non-principal ideal?

The ideal $R = (1)$ is always a principal ideal of the ring $R$, but $R$ is not necessarily a PID.

A principal ideal can certainly contain a non-principal ideal. Consider, for example, the ring of polynomials with integer coefficients. The ideal generated by $14$ and $7x$ is not principal, but it is contained in the principal ideal generated by $7$.