Can you have infection that does not show up in complete blood count?

Yes. A complete blood count only measures your number and type of white blood cells and red blood cells and platelets. A high white blood cell count can indicate an infection of some kind, but can never be used to tell what type of infection someone has.
Sure. Most bacterial infections will have an increase in WBC number, and type (segmented cells, pmns); sometimes may have either not both. Some viral infections have increase in lymphocytes, even with a low WBC count. Docs treat people, not lab tests-those are only part of the picture.
Yes. An elevated white blood cell count is a marker for an infection, however an infection can occur in the body and someone can still have a normal blood count.

Related Questions

How does a complete blood count show a possible infection?

Elevated neutrophil. Count. The total white blood cell/leukocyte count (TLC) and especially neutrophil/granulocyte count is usually elevated in patients with bacterial infections. Viral infections and whooping cough are associated with increase in lymphocyte count. Not all infections produce discernible changes in cbc. Read more...
Complete. This includes a differential white blood cell count showing the types of cells and their maturity. In bacterial (and some other) infections one may see immature white cells. Infections may also alter the red blood cells, the amount of hemoglobin, etc, and may also change the platelet counts, all of which are measured in a complete blood count. Read more...