Overtraining is definitely possible and really serious. I'm experiencing directly at the moment. Mavafanculo is spot-on with the symptoms. I had these, and I also started to get a constantly streaming nose, but also serious mood changes. I had been away from training for a few years, and wasn't taking anything - just wanted to get back into it. But I hit it way too hard, and it started with frequent sore throats and colds. Seen as I was always prone to these, I ignored it. My sleep was terrible, and I seemed to be making no improvements, but losing weight and gaining fat around my stomach. Six months in this, I upped my training thinking I was selling myself short, and suddenly improved. Two days later, my strength just dropped out of the sky, and then one morning I could barely conjure the energy to get out of bed, my eyelids were flickering like crazy, I had headaches and was dizzy all the time.
It's been 10 months now and I'm still not able to do any exercise at all, even light cardio can throw my back over the edge, and I stop sleeping for 5 days, etc. Dermacrine has helped balance the cortisol-testosterone ratio, and for the first time in a long time I can get some good sleep without waking up at 3am.
I don't think overtraining is purely about training, however; I was also suffering serious stress trying to make a career for myself, and doing serious travelling, including on my 'rest' day.
It's no fucking fun at all! I wonder if it is possible to recover if you get it bad!