I'm not aware of any studies looking at this. I think the three-year belief comes from clinical observations by doctors who treat us and anecdotal evidence in the patient community.
Myself and all the others I have spoken to became MORE ill the second time than the first and none of them then recovered again (although we remain hopeful)
That has certainly been my experience too.
I had mild ME with constant infections, mild dysautonomia, poor stamina and stress intolerance for about 10 years before the first serious relapse happened. In hindsight, I had mild PEM during all those years but I had no idea what it was, I just knew I didn't recover from exertion the way others did, and in fact I always felt worse for three days AFTER the exertion than on the day itself.
The first major relapse involved sudden development of POTS, subjective perception of constant hypoglycaemia and a worsening of existing symptoms which were subclinical prior to that. The relapse lasted about three years during which I was moderately affected. Sure enough, I "recovered" without any treatment whatsoever, just resting and staying mostly housebound. Around the time I "recovered" from this relapse I developed four chest infections in one winter (2006-2007) and took lots of antibiotics. That was the last time I had a real cold/flu.
For the following six years I was back to near-normal functional capacity in the sense that I could work etc. but I was still sick with poor stamina and tolerance of stress, as well as mild dysautonomia and lots of fibromyalgia pain (which responded well to exercise). I didn't have any noticeable PEM but my capacity to exercise was still greatly diminished compared to other people. I also had a peculiar inability to get normal illnesses when everyone else around me was sick. I would get a sudden feeling of severe illness every so often which would last a few hours, maybe half a day or a day, and I would think "oh, I'm coming down with something" but it never progressed to anything real. The only infections I continued to get during this period 24-48 hour GI viruses.
Three years ago almost exactly to the day I relapsed in a massive way with POTS, PEM made a comeback, and I developed host of new terrifying symptoms. In the year leading up to this relapse I had been eating a very low carb diet (big mistake), exercising too much (for me; normal people would have found my exercise regime laughable), and was having an unusual amount of stress and overexertion at work. I also got floxed with Cipro around that time for a supposed UTI which was actually IC. Anyway, like a FOOL I continued to push for the first six months of this relapse and was still going to the office a couple of days a week. Any attempts at exercise made me dramatically worse instead of better. My condition then took another turn for the worse in the spring of 2013 when I got an apparent enteroviral infection and I became instantly bedridden with another host of new very severe symptoms including severe generalised pain, mast cell activation syndrome, diabetes insipidus, episodes of massive hypertension superimposed on a baseline of very low BP and pulse pressure, POTS so severe even sitting was impossible for longer than 5 minutes, muscle fasciculations 24/7, frequent shingles outbreaks etc.
Anyway, I'm now at the three-year mark since the start of my last relapse in Aug/Sept 2012 and I get no impression whatsoever that there has been any tendency toward a spontaneous remission unlike during my first relapse 13 years ago. This time round, I get a sense that no meaningful recovery is possible (unless new treatments are developed targeting the underlying pathophysiology), and every bit of improvement I've made since the low ebb in 2012-2013 has been hard-earned via judicious use of various prebiotic interventions and ALA.
I personally don't believe full recovery from this illness (ME) is possible. I certainly think that full recovery is possible from garden-variety PVFS but those recoveries usually happen within 6-12 months of the triggering infection. I have spoken to a few people who thought they had recovered from "CFS". Objectively they did not seem recovered to me but had managed to convince themselves that a somewhat reduced functional capacity combined with avoidance behaviours (of triggers like overexertion, food intolerances, stress etc.) constituted recovery.