No, she can't read. But she does now recognize her own name, which I think is pretty awesome.
It started as just a habit that both Erich and I have. If Lily happens to have a crayon and paper (not very often yet as she doesn't know how not to make a mess), we will write out her name and say it. "L-I-L-Y. Lily!"
It didn't take too many repetitions before she approximated the same actions. "L! L! Lily!" she says with perfect little Ls as she swipes the pencil or crayon in vertical lines.
She has a book called The Parable of the Lily. Tonight, during her usual refusal-to-sleep routine (still ongoing, actually), I decided to test her. I pointed to her name in the title and asked her what is says. "L. I. Lily."
At least she's smartish. And pretty cute. :)