It's tragedy for parents to see their children die. To avoid that, children in Japan are told they shouldn't cut their fingernails at night.
It's dark at night, easy to mistakenly cut fingertips and bleed to death.
(A pun on the Japanese yotsume means cutting nails at night. A different kanji is also pronounced yotsume which means dying early, before one's parents.)
It would be most unusual to die from a cut finger. There are no major arteries or veins that run through the fingers, so unless you suffer from haemophilia, your blood will clot. (A different situation if you sever your finger completely, of course.)
There's a possibility of getting bacteria in your blood. Overreaction to the infection by your immune system could lead to sepsis, lower your blood pressure causing your organs to malfunction and be life threatening.
But looking again at the wording of the superstition; the phrase "you will die" is not the same as "you will have the most unlikely chance of dying".
The lesson for children is, of course, that they should be careful using cutting tools such as knives or scissors, whether it's fingernails or anything else. Children play and often mimic their parents and other people older than themselves. They'll probably see older people cutting food for cooking, cutting flowers, paper, cloth, hair, and all sorts of other things, much more frequently than cutting nails.