It's totally possible. If a nerve was nicked it could heal within days, weeks, months. It depends on the nerve and just can't be predicted. I have some nerve damage from a nick my back took during my first epidural over ten years ago. It shows up as a shooting, debilitating momentary pain across my back once a year or so, just enough to let me know it's still there. I spoke with the anesthesiologist who did my epidural and he said that sometimes, it just happens. Nerves are tricky and can regenerate when you don't expect them to and sometimes they take longer than expected.
If you notice loss of function that's a problem. In any case I'd call your dentist if you're worried.