I imagine right now you're probably saying to yourself "what?"
Well it's true. To a point. I do have my limits.
As I often tell my students, there is a difference between being hurt and injured. I can pop someone in the nose & hurt them and unless the nose "breaks" the person will most likely have intense pain, watery eyes, and possibly blurred vision. But a short time time later they are relatively okay. And injuries often the result of serious force and require a more lengthy healing phase. Another great example of this is professional athletes; you will often hear how they play hurt, but when injured they are taken off the field for a minimum amount of time.
While I have absolutely no problem training while hurt - and would encourage others to as well provided they take the reasonable precautions - I absolutely advise against training while injured. Injuries are, by nature, more serious and require the body to heal properly in order to avoid possibly re-injury. Can you train while injured? Yes. However, it is every dependent on the situation and requires an understanding of possible dangers and training limits from both the student, the instructor, and the class as a whole.
Many year ago, back when I was still somewhat new to the martial arts, I witness one of the guys in my class break his hand during a breaking demonstration at one of his promotions (side note: he passed). He ended up taking a few weeks off to let his hand heal, but returned two weeks later with his hand in a cast. He continued to train Kata and kihon, but withheld from kumite and other drills that might injury his hand.
Anyone that knows me can tell you I have issues with my knees due to multiple injuries over the course of my life. On a couple occasions I have had a cortisone injection and my orthopedic has ordered no exercise for the following week. However, when my knees are just sore, achy, or just acting up, my training continues. My stances just aren't as deep or pronounced as they should be. I take it easy for a class or two until they feel back to as normal as they can.
But while my stances are less pronounced, I utilize those classes to really focus on the other parts of my body. If my knee(s) are bothering me, I analyze the movements of my koshi and my arm(s) on every punch or block both leading up to the technique and during. And it is during those hypercritical self examinations I often notice my some of my own flaws in the techniques and make mental notes on what to work on.
Now when it comes to dealing with an illness, obviously if I'm ever sick to the point I should not leave my house or I think I might be contagious, I do not go. And yes, there have been a few where I have gone and had to leave early because I couldn't "tough it out" (for the lack of a better phrase). However, I cannot count the number of times I would have a headache and head to class anyways. And by the time the class(es) are over, my headache will be gone. While I have no scientific explanation behind it, I imagine the combination of my body releasing endorphins from the exercise and the simple fact of putting my mind on something other than the discomfort or throbbing helps rid my body of the pain. If I'm recovering from a cold and feel fine, but my body is achy, I'll participate in class the same was as if my knees were acting up - less physical activity but increased mental scrutiny over every technique.
Why do I train while sick or not feeling 100%? Because I love what I do. It's really that simple. If I'm not feeling well, train, and still not feeling I'm not out of anything except the a few hours which I would probably just fill watching Netflix, but I still got a training in. And Neflix will still be there when I get home.
But as always, know your own limit first.