"The Anatomy Lesson", a short story by Scott Sanders; first published in Isaac Asimov's Science Fiction Magazine, October 26, 1981, available at the Internet Archive; reprinted in several anthologies.
A college anatomy class preparing for an exam study the school's collection of human skeletons, but one who is somewhat late and gets the last skeleton finds that it is not quite human, actually.
The two persons who assembled the bones and are slowly turning into monsters are a medical student and a librarian at the Anatomy Library. The student, telling the story in the first person, checks out a box of bones:
Presently the librarian returned with a box. It was the size of an orange crate, wooden, dingy from age or dry rot. The metal clasps that held it shut were tarnished a sickly green. No wonder she wore the gloves.
There's something wrong with the bones:
Inside I found the usual wooden trays for bones, light as bird wings, but instead of the customary lining of vinyl they were covered with a metal the color of copper and the puttyish consistency of lead. Each bone fitted into its pocket of metal. Without consulting notes, I started confidently on the foot, joining tarsal to metatarsal. But it was soon evident that there were too many bones. Each one seemed a bit odd in shape, with an extra flange where none should be, or a socket at right angles to the orthodox position. The only way of accommodating all the bones was to assemble them into a seven-toed monstrosity, slightly longer than the foot of an adult male, phalanges all of the same length, with ankle-bones bearing the unmistakable nodes for—what? Wings? Flippers?
That's not all that's wrong with the skeleton. The student complains to the librarian:
Furious, I said: "It's not even a very good hoax. No one who knows the smallest scrap of anatomy would fall for it."
"Really?" she said, peeling the glove away from her wrist. I wanted to shout at her and then hurry away, before she could uncover that hand. But I was mesmerized by the slide of clotrh, the pinkish skin emerging. "I found it hard to believe myself, at first," she said, spreading the naked hand before me, palm up. I was relieved to count only five digits. But the fleshy heel was inflamed as if the bud of a new thumb were sprouting there.
A scar, I thought feverishly. Nothing awful.
Then she turned the hand over and displayed for me another palm. The fingers curled upward, then curled in the reverse direction, forming a cage of fingers on the counter.
[. . . .]
I tried to imagine her ankles affixed with wings, her head swollen like a double moon, her third eye glaring. "And what sort of creature will you be when you're—changed?"
"We'll just have to wait and see, won't we?"
"We?" I echoed, backing carefully over the linoleum.
"You've put the bones together, haven't you?"
I stared at my palms, then turned my hands over to examine the twitching skin where the knuckles should be.
The story ends there.