I'm always hesitant to say "impossible".
For instance, if you started with a simple, single-instrument MP3 recording, you could probably, with much effort, write a program that would analyze the frequency, duration and loudness of each note, and create the appropriate MIDI data to approximate it. It would almost surely be far more effort than it would be worth, but possible, sure.
Actually, it'd have to be not just a single-instrument, but a one-note-at-a-time instrument, like a horn or maybe a violin. Chords would be much harder to pick apart. Maybe you could do some really fancy fourier anaylsis and try to tease out the individual notes, but it'd be a serious headache.
So perhaps not actually IMPOSSIBLE, but surely FAR more trouble than its worth.