It could signify stimulating lactation.
It’s a little known fact that “Hands Across America,” that much-loved '80s fund-raiser, was inspired by an event hundreds of years earlier entitled “Nipples Across France.”
You know, it makes perfect sense that the sister whose nipple is being pinched has the Jack “Eraserhead” Nance hairdo, as if she’s saying, “Holy shit, what’s with the pinching?”
Remember the scene in Road Trip where the tourist girl tells Tom Greene that “Women don’t lounge around topless like that!” and Tom Greene replies “Oh yes, they do!”? I figure the artist was a Tom Greene type.
Heh, I knew what painting this would be about before I even opened the thread. 
I saw this painting in person a few years ago when I was at the Louvre. I even have a picture of it with my buddy standing in front of it and pretending to pinch the other nipple.
I have no idea what it means, but it sure is memorable! And isn’t that kind of the point of art? To make an impact on the viewer? Seems like this one worked.
Just my uninformed opinion. YMMV
It helps a little bit to realise that the painting was probably inspired by this earlier portrait of Diane de Poitiers, the mistress of Henri II, by François Clouet. It is also interesting to compare it to another portrait of de Poitiers. Note that she is touching her breasts and holding a ring. D’Estrées may have been deliberately appropriating iconography previously associated with de Poiters. (That it shows d’Estrées is uncertain - the Louvre calls it only Portrait présumé de Gabrielle d’Estrées et de sa soeur la duchesse de Villars.)
kjbrasda may well be right to suggest that the nipple-pinching signifies lactation. Pregnancy was a major career step for any royal mistress as it usually consolidated her position at court. It was just the sort of event that a mistress - or her lover - might want to use art to celebrate.
All you need to know is that the subjects, the painter and the location of the painting are all French! Damned cheese eating, wine swilling, nipple pinching surrender monkeys. Harumph!
Yes I have read this too, that this painting was inspired by the Clouet painting. There are several paintings from this time picturing a nude woman in her bath, with curtains surrounding the bath, and a servant in the background. So maybe this was a popular subject at the time. Although I have a hard time understanding why these women wanted their portraits painted in the bath! I know I wouldn’t 
As in the other aforementioned paintings, there is also a figure in the background, I think I have read that it is a servant or maid going about her work.
These ideas support a quote I found out in cyberspace: “It is assumed to be an allusion to the birth of César, son of Henry IV and his mistress, Gabrielle d’Estrées.” However, it sure is a weird way to signify pregnancy!
7 page analysis article re this painting in “Art History” for purchase.
Oh and this, courtesy of google translations 
The embrasure makes the spectacle by Pierre Fresnault-Deruelle
I read the translation, and I have no idea what it was trying to say!!
Her sister’s hair, right before her nipple was pinched, was laying down flat.
Simple. This was the winning entry in the “Baffle Art Historians of the Future” Contest.
A few more possibilities:
The curtains are definitely representative of a woman’s dress. Notice how the left side is lifted magically, with no visible means of support. This is the artist thinking of himself as God, able to reach out and change reality, without being seen. And that structure that they’re standing within is draped with a type of fabric that might be used for ladies’ undergarments.
This is obviously a sequel to another painting, lost to history. In the first painting, the left side of the “curtain” is down, covering the entire left figure except for the arm. (Notice how the right side is properly tied back in place.) The viewer would assume that the unseen left figure is a man, testing the child-bearing abilities of his intended. (Notice that she is holding the ring, but not wearing it.) Then in this painting, the left curtain is magically pulled aside, revealing the gender of the figure on the left. Sort of the 16th century version of “We’re here, we’re queer!”
But actually, I think there’s a whole different meaning in this painting. I think the artist was a woman and, in fact, a feminist; and this is her depiction of God creating Eve. Think of the Sistine Chapel’s depiction of God transmitting life into Adam through his fingertip. If God were female, transmitting life to Eve, what better place than her nipple? And Eve is holding the ring that Adam will eventually put on her finger (after he’s created from one of her ribs). And that figure in the background is sewing the clothes that they will have to wear after the encounter with the snake.
Also – doesn’t the whole parting-of-the-pink-curtain appear to be somewhat vaginal?
And what’s with the big TV in the background?
It was customary for artists to paint hands in awkward poses so they could demonstrate their knowledge of anatomy. That’s why you frequently see individual fingers pointing at strange angles. One of the most common hand poses used for this purpose was an exaggerated pinch. In this portrait this gesture is used twice; one of the women is pinching a ring, the other is pinching a nipple
It’s a depiction of a rehersal for the halftime show at Superbowl -CCCCLXXXVI.
Well, you know the old saying…
You can pick your nipples
and you can pick your sister’s nipples
but you can’t pick your sisters.
Interesting thread.
The only thing I have to add is that the sisters have nearly identical [powdered?] hairstyles, pearl earrings, blase expressions, plucked eyebrows, bodies, faces. The only differences I see are that Gabrielle’s sister has dark hair and is slightly shadowed. I think the picture is kinda creepy. The “matched set interchangeable robot clones” look contributes to this.
Isn’t there some school of thought that says that the only thing better than a threesome is a threesome with twins?
Oh, yeah, that’s hot.
I wondered whether the women are actually meant to be sisters, or different representations of the same woman. The woman in the background might be her as well. The painting could be a representation of her life. She starts as the woman in the background, clothed in respectability, working in secret. The nude woman on the left is her after becoming the king’s mistress. The woman on the right is her after becoming pregnant. The ring is both a trophy of her success and a symbol of how the child will solidify her relationship with the king.
Perhaps it’s a nipple ring they’re getting ready to insert?
perhaps it is a nipple ring and her sister is helping her with it. sometimes you need help with rings, tricky hinges and the ilk.
Yes. I believe it’s called “The Greatest School of Thought in the World.”