I've been experimenting with analysing vowels in Praat. Sometimes it shows a clear F4 (which seems to mean that F4 has a narrow bandwidth - I don't know whether amplitude comes into it as well) and sometimes not. The difference seems to be partly to do with nasalisation, but I suspect there is more to it than that - is there anything else that the presence or absence of a clear F4 says about how the vowel is produced?
Also, is it conceivable that the presence of a clear F4 could be part of the vowel specification? It strikes me that we may not be able to create an F4 resonance wherever we like, because the position of most of the moveable parts is dictated by the need to get F1 - F3 on target. If that's right, the frequency of the next resonator is going to depend on the shape and size of your individual vocal tract, rather than anything you can control.