Doctors often count the amount of neutrophils in a person's blood to see if he or she has a certain illness. Why would this be a useful test?

Because they will increase or decrease in specific diseases which, along with other cell counts, would help narrow down the illness.

Explanation:

Lymphocytes one of the subtypes of immune cells present mainly in the lymph of the lymphatic system. They are composed of 3 types of cells

T-Cells

B-Cells

Natural-Killer (NK) Cells

T-Cells and B-Cells function as detection cells. Where they recognize & mark "non-self cells" or "infected self cells" as potential threat to the body and initiates or signals for their immediate elimination.

NK cells on the other hand, will recognize & eliminate "non-self cells" or "infected self cells" directly.

These cells will mostly respond to viral infections or tumor cells. Thus, their blood levels will increase mostly in these situations.

This means that if the patient is suffering from a fever or a sore throat, then the doctor can conclude whether this infection is caused by a viral or bacterial infection depending on how high Lymphocytes increases in the CBC test.

However, it should be noted that it is not used as a "sole" test but one of the tests to help narrow-down and pin-point the illness.

I hope that answered your question.