Everything I've read says no.  However, a neighbor of mine invited me to his front yard yesterday to pick figs from his two, huge Celeste trees.  He has been picking tons of figs for several days now, and had some to share. 

He preferred slightly less ripe ones because he was going to make preserves, so he offered me the ripest ones.  I was in Hog Heaven.  I picked the softest, hangiest ones, enjoyed some right on the spot.  Took a couple of dozen home, shared some with my husband, froze some and experimented with two of them.  I left them on the counter overnight.  This morning, they were not only MUCH softer, but when I tasted them I nearly fainted.  They were tons figgier, and sweeter than the ones which I ate the night before, which appeared to be exactly the same ripeness at picking.  They were the best things I've ever flopped a lip over.

What's up with THAT? Is the material I've been reading simply wrong, have I misread it, or did those two trees just not read the same book?

Thanks!

They will soften more if you let them sit a day, perhaps there is minor residual ripening, but it's more likely that you just had a clean palate to start the day.

They will soften a bit, but they will n ot develop more sweetness or flav or. If you actually want to test it, you wouod need to eat 1/2 and then wait a day and eat the other half. Otherwise yhou would be eating 2 different figs, one of which might have been a bit more ripe to begin with.

It is possible that some water evaporated leaving the fig flavor components more concentrated.  All of the above probably contributed some.