Can I file a file a lawsuit against an orthodontist who refuses to reimburse for services not rendered?

It all depends how your arrangements were set up.  In orthodontics, the bulk of our cost is at the very beginning of treatment….x-rays, models, photos, diagnostic time, the braces and other appliances, and time putting on braces.  If your payments were set up with a significant initial payment of perhaps 20% of the overall fee (ex. $1000-1500), then you are essentially paying for treatment as it goes.   However if there was no large initial fee and payments are flat all the way through, your doctor has some catching up to do to recover the initial up front costs.  After reading this you feel you are still not being treated fairly, I would ask them to review with you in person how they came up with the amount they feel you are due.  If you are not happy with it, then a formal letter to the doctor is appropriate.  Suing would not be worth the time and cost it takes to undertake it.  I have other suggestions, but these are the first steps I'd take.