As Sally Haslanger writes, proper answers to your question do need to feed on some empirical, in this case historical data. But I suggest that these data can only yield answers through some philosophical unpacking.
The category of mental illness is itself a cultural product, insofar as it depends on the notion that 1) our minds can be ill just as our bodies can; 2) and so that they are instantiated, most probably, by the brain. Note, however, that to restate the concept of mental illness in such terms is not to say that it is merely a cultural product, that truths are relative to cultures, that if some people believe our mind is instantiated by our feet then so be it. I say this because the kernel of your question seems to address the matter of whether it is useful to imagine a clear-cut distinction between language use - that is, broadly, culture - and a "substantive" reality.
To this I would answer No: what is described by the expression "mental illness" must correspond to a reality which must always have existed. To note the historical genesis of a concept is not to deny the existence of a reality designated by that very concept, by those cultural terms. Historians can go beyond older uses of language to tell us how that reality was perceived, understood, conceptualized. What we need to remember, then, is that culturally produced terms are not exhaustive of their conceptual object. "Mental illness" designates something real, but that reality can only be named in the way that it is at any given time, via the existence of further cultural constructs.
"Depression" is a relatively novel term; but it designates a number of conditions that doctors and philosophers had recognized even in Antiquity. One notion to which "depression" corresponds is that of "melancholy", an infamous term, central to western culture, believed from the 5th century BC to be the counterpart of creativity, inspiration, insight, and so on, but also of sadness, despondency, love-sickness, and madness in various guises. Its history has been told many times, in books, compilations, most recently an exhibition in Paris called Mélancolie: Génie et Folie en Occident. Melancholy has a much broader range of meanings than does depression: it is a category that encompasses states of extreme disturbance, from what we call mania to what we call schizophrenia and beyond, to the gentler sort of everyday "blues" all of us experience. Only a serious melancholic - someone who could not function in society - would have been considered in need of medical care (and medical care was not very gentle). A case of mild melancholy was nothing dramatic.
Literally, melancholy is Greek for "black bile", one of the four humours that corresponded to temperaments, to states of health and illness from the Greeks for two millennia on. But its meaning, significance and aetiology gradually ceased to be tied to that very precise physical substratum. (You could browse Robert Burton's Anatomy of Melancholy, published in the 1620s, to get a profound sense of the concept's fate.) In that sense, the term "melancholy" remained a cultural container in the same way that "depression" is today: we might assume that depression is aetiologically reducible to neurotransmitters, whose explanatory structure is akin to that of a humour like black bile. Yet there must exist cultural conditions for attaching a meaning or value to this putative reducibility: it isn't a stable given. In fact, the sliding scale between full-on madness and bluesiness, once covered by the concept of melancholy, no longer exists: today we must be either healthily content or depressed.
And so, one hypothesis for the rise in diagnoses of "mental illness" is that we tend not to accept natural states, which can be unpleasant: instead we are quick to turn them into pathologies that then justify the resort to medication. It is hard to gauge whether there has been a rise in cases of evident psychoses, or in cases of manic depression and so on, all of which have always historically been perceived as illnesses that necessitated a cure - and one can only be grateful today for the medications that do help those in acute distress. But I imagine far more people are diagnosed as "mentally ill" than is warranted. We like short-cuts to well-being, easy answers to painful questions, so we pop pills. We forget that no one quite knows the criteria, other than those provided by symptoms themselves (categorized in the DSM), by which one can define the border between a mentally ill patient in need of medication and someone suffering from common melancholy, in need of introspection and consolation. And we lose sight that we make a fallacious inference when we say that, if our mind is our brain, then all mental distress is simply a medical ailment. A hernia, a collapsed lung or a broken leg are clearly medical issues; but mental matters, that is, matters of psychology, are less obviously unpacked by the appeal to medicine. (Incidentally, psychology is a word that used to denote what its etymology indicates: knowledge, or for us science, of the soul, or for us mind.)
Many doctors in the past used to be philosophers, precisely because it is so hard to establish what exactly is the province of medicine; and today, good doctors and good psychiatrists still know that diagnosis is in part a philosophical act.