if you're in a bathtub with warm water the blood doesn't coagulate, so theoretically you could bleed out from a papercut. I pulled that papercut part out of my ass but the first part's (mostly) true
It's not completely true. As I've studied hemostasis depends on many things, and temperature is none of them. That might be an urban legend. What I've read about the warm bath is that helps to calm the pain.