What problems can occur if I floss too deeply?

Incorrect Flossing. Improper flossing can cause unnecessary injury to the teeth. The floss needs to wrap around the tooth, not pushed deeply between them. Consult with yours dentist or hygienist for proper technique.
Bleeding, receding. Bleeding, for one. But if you make a habit of flossing to deeply then over time you could cause your gums to recede between your teeth.