If an organ was to start failing, not total failure, such as the thyroid gland, is it possible that the fault may not lie in the failure of the organ but in the nerve signals coming from the brain?
In many hormonrs, the hypothalamus produces a hormone which acts on the pituitary gland which subsequently acts on an end organ. Taking your example of the thyroid gland let's take a look at it.
The hypothalamus produces TRH which acts on the anterior pituitary causing it to release TSH which in turn stimulates the thyroid gland to make thyroid hormones T3 and T4.
Abnormal levels of thyroid hormone may be caused by problems in any one of the three stages. Problems on the end organ are termed primary, in the pituitary gland are termed secondary and in the hypothalamus tertiary.
For simplicity let's look at abnormally high levels of thyroid hormones called hyperthyroidism. In a patient with a primary or thyroid gland problem we expect high levels of thyroid hormones but low levels of TSH. The pituitary gland is trying its best to tell the thyroid gland, "No more!" but it won't listen. In secondary or pituitary disease, the pituitary gland is the bully pushing the thyroid gland to make lots of thyroid hormone by over production of TSH. Usually there are other hormone disturbances too in this case as the pituitary gland has a range of hormones it secretes. Also the swollen pituitary gland puts pressure on the nerves carrying vision leading to visual loss (specifically in the temporal fields). In this case the hypothalamus will be secreting low levels of TRH (which isn't routinely measured) trying to get the pituitary gland to stop acting out. The rarest situation is when the hypothalamus has the problem where every hormone is increased. However again this will be causes by something or the other detectable usually on imaging.
So the short answer is yes, but there will be other features. Primary or end organ dysfunction is most common and will cause characteristic findings in blood tests and examination (e.g. swollen thyroid gland) but secondary and tertiary disease can be figured out when other features are found in association.