This excellent scientific article by Chatzitomaris et al discusses the various NON-thyroidal causes of thyroid hormone T4, T3 and pituitary TSH imbalance.
Chatzitomaris, A., Hoermann, R., Midgley, J. E., Hering, S., Urban, A., Dietrich, B., … Dietrich, J. W. (2017). Thyroid Allostasis–Adaptive Responses of Thyrotropic Feedback Control to Conditions of Strain, Stress, and Developmental Programming. Frontiers in Endocrinology, 8. ARTICLE LINK: https://www.frontiersin.org/articles/10.3389/fendo.2017.00163/full
The conditions that lead to low thyroid hormones (and lower TSH despite low thyroid hormones) can also affect hypothyroid patients on thyroid hormone therapy, but if they are taking Synthroid / Levothyroxine, the medication may keep their T4 high-normal while T3 can still go low.
In a hypothyroid patient on therapy, if Low T3 is not caused by any of these non-thyroidal health factors, or T3 stays low when these factors resolve, or if T3 stays chronically low and prevents recovery from illness, then — either their dosage is not high enough, or T3 hormone may be needed in their medication.
These other physiological influences on thyroid hormones make it necessary for doctors to have good training in the interpretation of thyroid lab test results. Thyroid hormones are not simple– they affect and are affected by the whole body in complicated ways.
It also means that testing Free T4, Free T3, and Reverse T3 has other uses — it can aid in confirming the diagnosis of other types of illness or physiological conditions, and it can explain hypothyroid symptoms in many patients who don’t even have a problem with their thyroid gland.