I have a tree purchased from Park Seed as a "Petit Negronne" dwarf.  It is about 7 feet tall with good structure, and it gets lots of very dark figs (early and late) but they are pulpy and completely inedible.  My other figs (Brown Turkey, Celeste) in a different part of the yard, do yield edible fruit.  I have tried lime, fertilizer, more watering, etc., but I have never had a single edible fruit from this tree.  Any advice, or should I cut this down and start over?  I am located in PG County Maryland, near Washington DC.  Thanks for any help or advice.

Just a wild guess but maybe it is over loaded with figs and just not enough energy to push them to ripen?

Try taking a third or half of the newer figs off and let the older ones ripen. You can also clip the bud tips of the branches off too.

I had a Negronne long ago and it too didn't ripen. I can relate to your frustration. Had I only known then what I know now.

Others with more know how should chime in soon.

Good luck

wondering if maybe you have a capri fig. I purchased Petite Negra from Parks which was later id'd as Aubiqua Petite. It is possible their supplier mislableled.

Thanks folks.  I have eaten figs from lots of different trees, and though some are better than others, I had never encountered a seemingly ripe, purple fig that had no moisture or no sweetness at all.  That's why I am wondering if there are just some varieties that are more ornamental than edible in my zone, or if there are some special cultivation techniques I need to try on this tree.

Pictures might help identify it as a capri or not. 

Most Likely is a male,made from seeds,if you find dust like matter inside fruits it is a male.
Maybe Parks seeds, gets plants from a whole seller,or more than one,and is very hard to track down,who did what.
But ,there is no excuse why the fruits are not edible in Maryland.
And if  not all get ripe because of short Summer at least a handful that appear first on branch should be edible and tasty,but in your case,they are not,so I suspect,someone was playing with seeds,and that is how your tree came to be!!!!!.