Hi Chauncey,
I agree that if your p-levels remain below 9 during an single pass adaptive (SPA) analysis, then the solution is reasonably numerically accurate. Of course, it's impossible to make precise statements about accuracy because so much depends on the specifics of the model and of the analysis.
As you probably already know, you can get an idea of the accuracy of your results by looking at the RMS stress error estimate that is printed in the summary file. You can also create a result window that shows the distribution of stress error estimates. You can check the accuracy of the default SPA analysis by running a similar SPA analysis or multi-pass adaptive (MPA) analysis with tighter convergence criteria.
I don't think there is a "mathematical sweet spot" or rule of thumb that balances the number of elements with the p-levels of the edges for all problems. I do know that, in general, increasing the polynomial order of the edges increases the accuracy of the solution faster than increasing the number of elements does, at least when measuring error in the energy norm versus the number of degrees of freedom. (See, for example, Chapter 8 of Joseph Flaherty's course notes on Finite Element Analysis.) However, singularities and stress concentrations introduce exceptions to this general rule, and that's why adding smaller elements near fillets and isolating singularities often help to obtain accurate solutions faster. In another post, Jonathan Hodgson mentioned the Edge Length by Curvature AutoGEM control, and I agree that it's an excellent tool for helping to create good meshes.
Tad Doxsee
PTC, R&D