I did almost the same thing with an X-acto knife once. Up late working on a model for an architecture final, was tired and punchy, not being careful; the blade was either dull or got caught on something on my mat and skipped out of the foamcore I was cutting, across the tip of my right index finger that was holding the board down.
I cut off from the middle of the tip of my finger, down the same side as you to about where the edge of the nail meets the cuticle. It really stung, so for some reason I started flailing my hand around... that project ended up looking like a crime scene.
Aside from that part of the nail nothing else was fully cut off, so 5 stitches and some fun medicine were all I got. Had I looked down and seen skeleton fingers sticking out, I probably would have passed out.