Yes, your deductible probably has not been used yet; $80 a session for therapy or a medical visit does not sound too bad to me and getting both for that price would be acceptible to me. But if you are being double billed and don't like it, I'd change your type of therapist to one who is not also medical-related, a social worker or something.
Here's a good definition (Oregon's) of what a NP does; the "counselling" in this list is medical counselling, how your meds are affecting you, for example, your side effects, so that is not therapy, psychotherapy is not the same kind of counselling. If you have medical conditions so that taking meds is a mine field (high blood pressure meds, heart problems, diabetes, etc.) I'd stay with the nurse practitioner as you can more quickly get help than just through a psychiatrist; I'd see it as a boon that you were seeing someone with both medical and therapy knowledge.
http://www.medscape.com/viewarticle/422935_5
Or, you could see if your nurse practitioner could write your psych scripts if you are "stable" and just doing maintenance, and then you would not have to pay a psychiatrist, too.
No one "has" to do this, they are allowed to and they (clinics/practices) need the money so they are. Talk it over with your NP and decide if you want to come to some other arrangement so they aren't getting 3 payments, only two (and/or you could have your regular medical doctor prescribe your meds if you know what works and s/he's okay with it).