Yes, iron and/or nutrient deficiency can cause hair loss, but not miniaturization of hair. Miniaturization happens only in Androgenetic Alopecia, when DHT shrinks the hair follicles.
My cousin is anemic and has iron deficiency. When I saw the top of her head today, the thinning looked exactly like she has Androgenetic Alopecia going on. Maybe it's totally unrelated but it can't hurt to draw a blood sample and check it out.
Miniaturization is most commonly seen in Androgenetic Alopecia, but it is not the only thing to cause it. The following are causes of miniaturization: Genetics (Androgenetic Alopecia), DHT (the amt of which is also determined in part by genetics), aging, telogen effluvium (which can be caused by stress, physical trauma, prescription medications, nutritional deficiency), or nutritional deficiency alone.