Although many textbooks botch the job, the sick euthyroid syndrome isn't really that difficult a concept.
In any serious illness (acute or chronic), the body down-regulates thyroid function. Specifically, it reduces TSH secretion, and inhibits the conversion of T4 to T3 (preferring to convert it to the inactive rT3).
The effect of this is that T3, TSH and T4 may all be low during a serious illness (the latter secondary to the low TSH). If you aren't aware of this possibility, you may interpret these results as those of hypothyroidism, probably secondary to pituitary dysfunction.
The way around this is twofold. Firstly, take a history, and while you're at it, examine the damn patient. If (s)he shows physical signs or symptoms of hypothyroidism, or a pituitary tumour, for instance, then it is much more likely to actually be hypothyroidism. Secondarily, if in doubt, simply repeat the thyroid function tests once the patient is better. This will rule out the possibility of the 'sick euthyroid syndrome'.
P.S. What is the body trying to achieve by down-regulating thyroid function during illness? I'm not sure whether anyone has a better idea, but to me it seems a logical thing to do. We know that the T3 and T4 are major determinants of the basal metabolic rate. Furthermore, we know that in serious illnesses, the body's metabolic rate is also raised - by up to about 100%. So, to keep us from burning up completely, perhaps the body chooses to lower the secretion of the thyroid hormones...?