Yes, it is possible to have both Hashi's and Graves.
Make sure you insist on getting TGab done as well as the TPOab and TSI, if you've never been tested for antibodies before. TGab and TPOab, both indicate Hashi's and some people have one or the other of the antibodies, while others have both; therefore, Hashi's can not be ruled out without both tests.
I agree with Barb135 but would like to add that if you are having a really hard time getting your thyroid levels stable then running TPO and TSI might not be a bad idea. I am actually getting set to have those run soon as apparently it is possible to have Graves AND Hashimotos. If things are stable though then I would think the thyroid panel would be good enough.
My endo continues to test for TPOab, every time I have blood work, but no, it's not really necessary. Once you have an autoimmune, you always have it. I think some doctors just like to keep track of the antibody count.
DO make sure the T3 and T4 are FREE, not total...... TSH, FT3 and FT4 are sufficient.