The miracle of importance sampling is that it doesn't formally ever prevent you from visiting the rare regions, it just might take longer. What can occur is tha

The miracle of importance sampling is that it doesn't formally ever prevent you from visiting the rare regions, it just might take longer. What can occur is that you will get what appears to be a well-converged estimate of an answer but really you weren't fully ergodic. Such is the curse of dimensionality. No free lunch theorem, yadda yadda.