Nope, Still Alice is not based on a true story - but rather a fictional book of the same name.
However, the author of the book, Lisa Genoa, is a neuroscientist with a PhD from Harvard University.
Her experience in this field, and the time spent working with real life sufferers of degenerative diseases means that she was able to make the story very realistic.
It's likely that the people she met and worked with influenced her work and probably feature in some way - so you could argue that their is an element of true-story to the plot of Still Alice.
Genoa also spent years working and researching at Massachusetts General Hospital East, Yale Medical School, McLean Hospital, and the National Institutes of Health.
She also taught neuroanatomy at Harvard Medical School.
Here's the trailer of the Still Alice movie in case you haven't seen it yet: