At some point, getting sick is going to get in the way of your running. The usual advice is that it’s OK to keep running if your symptoms are from the neck up (runny nose, woozy head, etc.), but that you shouldn’t run if your symptoms are below your head (congestion in chest, coughing, vomiting, etc.).
Like so much in running, however, running when you’re sick is a judgment call, not a black-and-white matter. I almost always at least attempt a token jog, on the theory that half an hour after the run I might feel significantly better than I did before. The run is unlikely to be a peak experience, but it’s something different than sitting around with a box of tissues, and I usually feel a little cleared out from the increased blood flow. I stay close to home in case it becomes obvious 10 minutes into the run that further escapades are contraindicated. But even on those occasions, it’s hard to argue that a short jog was a mistake—it’s not like doing so will have made you that much sicker or delayed your recovery.
Again, this is a judgment call. Some of us just like to run pretty much every day. I like the thought that the cold or flu, which is already lowering my quality of life, didn’t completely upend things.