Is it possible to have cancer and normal blood count at the same time?
Cancer doesn't have to have a profound effect on the blood. Remember that a cancer generally starts off as an abnormal tissue growth or a neoplasm. As the tumor progresses into a malignancy, it has the potential to gain or lose functions, develop new cell types that shouldn't be present in a given organ, invade other tissues, etc. Because of this, in some cases of cancer patients may be subject to paraneoplastic syndrome, or cross-reactivity between something the cancer is doing and the normal tissues. 

In the above-linked source, some hematologic paraneoplastic syndromes include (1) eosinophilia, commonly associated with a number of cancers, (2) granulocytosis, often associated w/ lung cancer, (3) Pure Red Cell Aplasia, commonly associated w/ thymomas. The list goes on, but in some cases blood testing is capable of assisting in diagnosis, especially in cases where cancer of the blood is the culprit (very obviously). To also be kept in mind, they note paraneoplastic syndrome only affects around 8% of all cancer patients.

In other cases, you'll be monitoring treatment. As we know, chemotherapy is destructive, and can cause a depletion of platelets or neutrophils, damage organs, etc. Complete blood count (CBC) testing will help oncologists monitor treatments in cancer patients.

Being said: Because individuals with a tumor/cancer may fall within normal ranges in a CBC test, and because these tests must be analyzed scrupulously due to several factors that influence them such as diet, multiple tests are often needed to arrive at a meaningful conclusion. On the basis of cancer itself, however, if it's a solid mass lodged in the liver for example, it's completely possible that blood counts are normal. What you may find, however, are biomarkers in the blood that are saying "hey, the liver is being damaged or dying" such as the enzyme aspartate transaminase. If the tumors are currently small, you may not find anything meaningful in blood tests, or that the blood is "normal." (the possibilities aren't exhaustive)

As a matter of real data I was able to find, Spell et al. was at least able to find some efficacy in CBC testing for colon cancer, to 84% sensitivity and 88% specificity, but you should suspect that individuals may still be asymptomatic under such tests regardless.