What are the chances of getting sepsis from bitting skin around nails?

Rare: Slim chances. When you say sepsis, to me that means the very ill hypotensive state of shock one gets with a serious bacterial infection. In my almost 20 years of seeing patients I have never seen this result from biting nails around skin. Now, getting a local cellulitis, yes, that happens often. But you won't get septic with no warning, in other words, local infection will come first.