Pain between teeth - is there something stuck?

[howdidyougethere]

I have a horrible habit of biting my nails, and I ended up having tiny pieces of nail circulating in my mouth. At the same time, I started getting a pain between two of my teeth. Therefore, I thought I'd gotten a piece of nail stuck, and attempted to remove it with floss and even thin cotton. After no show from a piece of nail, I started to see blood on the floss and cotton, and stopped, fearing further injury. The only other possibility was that I'd cut myself between my teeth slightly. It does not feel like something between the teeth but - if there were something stuck - it would have to be pushed up into my gum.
Resting my jaws naturally together doesn't hurt, but any sort of pressure hurts like a piercing pain.
Do you think this is a piece of nail stuck in my gums, or a cut from the cotton/nail I bit off?
Thanks x