The initial wound pinches yes, but if the blade is sharp enough it cuts through quite easily, like a scalpel would not hurt as much as a knife. I find the release of blood to be rather interesting and I find my concentration goes up afterwards, perhaps as a result of a traumatic event.
I have not experienced a life threatening cut however, that may be a lot different, but it would be interesting to test if there was a way to do it safely. Chop an arm off with an axe and have a surgical team sew it back on right away or something.
I read that death from loss of blood is painless, aside from the wound damage of course.
Hurting myself doesn't affect me emotionally, and I don't hurt myself when I am upset either, which also invalidates the above comments I suppose.