Thanks so much for asking this question Jacqueline, and many thanks to Greg for the informative answers. I'm wondering if this name might be associated with a similar name I've been researching that seems to vary between Caille, Cailler and Caillet? The name is of great interest to me because I think it might be the origin of the surname of one of my more challenging ancestors Exilda Cady. All of the records I find for her use the name Cady so I was very surprised when I took a trip to the cemetery where she is buried to clean up her grave marker, a photo of which had been sent to me by a DNA match of my mother's. Unfortunately the marker had been overgrown and her surname was ever so frustratingly obscured by lichen, though you could see the "CA". I fully expected it to be Cady so imagine my surprise when it turned out to be "Cayer". This actually made more sense since most of the sources seemed to indicate that both her parents were born in "French Canada" and the Cady name seems to have originated primarily in England and come to America through an early settler at Plymouth Colony in MA. It also turns out that some of her siblings used the name Cayer on occasion, but oddly I find no records of her father ever using the name.
This led to another discovery when I found that a DNA match was descended from Olive Currier who it so happens also used the name Cayer on her grave marker. I have not yet been able to make the connection but she appears to be our most likely link at this point.
So it seems that the name Caille (and perhaps Chaille as well) may have been anglicized to Cayer and then eventually to Cady and Currier in some cases. I would be very interested to know if anyone else has seen any occurrences of this evolution.
I've also recently worked on some of the Caille's in Canada (descendants of Jean Caille who was born in France and emigrated to New France) and discovered one in Louisiana, Susanne Caillet who is an ancestor of another DNA match. Does it seem reasonable to suspect a connection between Jean Caille and Marie Magdeleine Chaillier?
And finally, FWIW, I did a quick check and my Mom has a very small DNA match (5.5 cM) with Kathleen LeBlanc who is descended from Marie Magdeleine Chaillier. I would normally ignore a match this small but it is very interesting because it overlaps a match with a descendant of Exilda Cady's brother Dennis Cady.