surfsatwerk is just purposefully being obtuse about the most common meaning of the term 'overtraining' when applied to exercise. Follow a nice progressive increase, stick with it, and yes, you can train 6-7 days a week and multiple times day. However people quite commonly push themselves beyond what their body is currently able to recover from given their level of fitness and the amount of sleep/food they get, and they call it overtraining.
A good exercise regimen will generally improve your immune system in the long run. A really tiring workout will tax your immune system in the short run. But yes, even if you're improving your immune system in the long run by increasing your fitness level, you're still susceptable to the germs everyone else is bringing in, so be smart about that.