Window period. If a person has been recently infected and has not had time to make antibodies to the virus, one may test negative by the usual screening tests that look for antibodies to the virus. If you believe that you have been exposed to HIV and test negative, it would be advisable to get tested again after about six weeks.
Asymptomatic HIV. Hiv infection for the most part is asymptomatic. A small percentage of individuals who are initially infected with HIV will develop acute retroviral syndrome ( a mononucleosis-like syndrome with fevers, rash, lymphadenopathy) while another percentage will develop some flu-like symptoms. Asymptomatic HIV is usually detected at a doctor's visit for other reasons (eg. While donating blood).
If recently acquired. It is advisable to get retested if risk factors are present or to confirm negative antibody status..
Yes. You can have HIV and have a false negative (wrongly negative) test. There is also a "window period" between the initial infection and when the test will pick it up. Consult your doctor.