This may sound crazy, but sometimes I get weird cuts on my fingers that are deeper than a papercut but never bleed. I figured out it was from my razor in the shower. WHich is why it never bleeds, I don't feel it because its not that deep and I am busy doing other things so I don't notice!
I guess I clean it with my fingers and it just goes the wrong way.....
I know they make something for this called Crack Cream or something like that. It's when your hands or fingertips are so dry the skin just kind of cracks. I've had these before and it's usually brought on by a change in weather (usually the colder weather). The cream really works, but if you go to a Natural food store they sell straight lanolin in a tube and that works REALLY well. I used to put that on before bed time, then put a layer of Bag Balm over the Lanolin, then wrap my hands up in saran wrap or cotton gloves. I used to have REALLY dry hands as if you couldn't tell. Let me know if you need more info!