I worry less about ring count and more about wood density and the spring wood to summer/fall wood ratio. I do notice that within a species, there is a modestly strong positive correlation between density and ring count per inch.
Spring wood is the highly porous wood that forms the lighter boundaries between rings. It is relatively weak. The thinner the spring wood sections of each ring, the better. I have also noticed that trees that are towards the northern extent of their range (or harvested at high altitude) tend to have thinner spring growth relative to summer/fall. I tend to harvest species that are near the northern extent of their range in my area.
I just harvested a bitterroot hickory blowdown that had 12-16 rings per inch, but the spring growth was so thin you can hardly tell one ring from the next. It is drying slowly but is incredibly dense and will make great bows and arrow shafts.