I apologise for repetition if this has already been highlighted. It related to a study carried out in Amsterdam:
If you had a TT for Graves' then any Thyroid Stimulating Immunoglobulin remaining in the blood will possibly affect the TSH levels by fixing on TSH receptors in the pituitary. There is a study published by Mark F Prummel and Leon J S Brokken in the Netherlands which tested this hypothesis, first of all in rats and then in drug-treated Graves' patients, and came to a conclusion that this is, indeed, the case.:
This study did not address the situation of TT-treated Graves' sufferers, but what it did do was demonstrate the existence of TSH Receptors in the Pituitary and the effect of Thyroid Stimulating Immunoglobulin on the TSH levels.
If a Graves' patient appears to have a 'suppressed' TSH and yet feels under-medicated, then they would be advised to test for TSI. If there is still TSI in the blood, then testing for T4 and T3 would be one way to establish if the patient is on the correct medication - plus observation of their symptoms.