I think this is a common question for runners and most people answer with mention about the above the neck and under the neck rule and so on.
Well, firstly it obviously depends what you are sick with. A bad stomach can certainly down many people while a cold can usually be run with.
I think it is different for every runner. I read an article this morning about how do you know when you are sick enough to stay off work. I loved the answer as it is exactly the same as my answer for the question of how do you know you are too sick to run.
The answer: If you are wondering then you are probably OK to run (or go to work). If not you will know that you just can't. That's how I know when not to run when I'm sick. I just know I can't. If I wonder, I will go out for a short, easy run and see how it goes.
This past two weeks have been very frustrating as I've been battling with especially a sore, burny chest. An irritating cough accompanied this and I just knew I could not run at all. I am feeling ready to run today for the first time and will go for an easy 5km tomorrow.
All part of the fun I suppose...
Happy running everyone!