Can a complete blood count tell you if you have mono?

No, but. A blood count can tell if you have an abnormal counts of the cells of the blood (white cells, red cells and platelets). One or all can be lower with mono. However, even if you have an abnormal blood count and the right symptoms, you should have the monospot test done to confirm.
No. A complete blood count only measures the number and type of white blood cells and red blood cells. There are very specific blood tests for mononucleosis.
Kinda. CBC shows number and type of white blood cells. Mono has "atypical" lymphocytes (type of wbc). Can be confirmatory in appropriate setting, i.e. Fatigue, sore throat, enlarged lymph nodes, (especially behind the neck), achy, feverish, with or without weight loss, poor appetite, jaundice; can (should) be confirmed with more specific tests for epstein barr virus and/or cytomegalovirus (causes).

Related Questions

For how long after I have mono will my blood count of monocytes be elevated?

It's not. Monocytes are not elevated in mononucleosis. The cells in the blood that indicate mono are atypical lymphocytes, and they usually disappear within a few weeks.
Not important. Is your diagnosis of infectious mono correct? Because high monocytes can represent a variety of other diseases also. If you had lab proven infectious mono, the monocytes stay up for 4-8 weeks but that is not important. If you are worried re infecting others, the + igm of ebv is what determines if you can infect someone else and that may stay up up to 18 months.

Can you explain if it's possible to have mono with a normal CBC blood test?

Yes it is possible. The best way to determine if you have mono is to do a monospot test or do a blood test for the epstein barr virus igm titer. If positive it would tell you that you have mononucleosis.