What Causes Productive Cough Along With Vomiting With Negative Strep Test?

Can a strep test come back negative I correctly? My daughter and son got sick around the same time. Both had a red ring in their throat. My dtr was diagnosed with strep and started on an antibiotic but my sons test came back negative. My dtr is getting better but my son is coughing productively and now vomiting. Could his strep test have been wrong? They sent it to the lab and it came back negative.