Where I've gone wrong in the past: too many consecutive days of running. Granted, the type of running matters too, and in what context. For example, running 5 times a week when the past 6 months have been sedentary. Bad idea.
I was never a coached runner, but I've read and heard about "workouts" which from what I understand generally refers to the non-junk miles days where you actually focus on a specific pace. Don't do two days of workouts back to back. That's where recovery days come in. Save those rest days or easy pace runs for the day after a workout (in my case this is either intervals, a long run, or a full-effort 5-10K. It's also a good time to do non-running fitness activities such as core, weight training, poodle grooming, and the like.
Gradually increasing mileage and number of days run is also sage advice. I've seen percentages thrown around, but that seems really arbitrary to me. The number of miles probably doesn't matter as much as how they are run. At any rate jumping from 10 mile weeks to 30 mile weeks is asking for trouble.
What happens if you overtrain? From my experience-injuries. I've experienced 3 major injuries that all came down to the "too much too soon" principle. Shin splints-running in Vibram Five Fingers 20 miles a week on pavement. Achilles tendonitis-probably from amping up mileage in zero drop shoes too quickly. Patellofemoral pain-running with weak legs. This one used to really scare me. Since I've strengthened my legs, I have very little knee pain. Leg strength takes time and for me lots of lunges, squats, and using the stairs all day at work.