I'm reading about variational principle in quantum mechanics, which basically states that smallest (algebraically) eigenvalue $E_0$ of an Hermitian operator $H$ can be calculated as the minimum of the expression
$$\tilde E_0=\frac{\left\langle\psi\right|H\left|\psi\right\rangle}{\langle\psi\mid\psi\rangle}$$
over the space of test functions $\psi$ in the domain of $H$. But there is a caveat, as said on this page:
There is no a priori guarantee whatsoever that the trial state ${|\psi(\mathbf{x}_*)\rangle}$ corresponding to the minima of the energy expectation value ${E(\mathbf{x})}$ at ${\mathbf{x}_*}$ has any resemblance to the ground state.
I wonder, how far off can such a trial state be, if it gives some high precision estimate of the eigenvalue? What is an example of a really "bad" case where the test function is very different from the true eigenfunction, but still estimates the eigenvalue very precisely (i.e. eigenvalue is correct up to $N$ decimal places while test function's precision as estimate of eigenfunction is much worse)?