Yes, it is possible... If there were periods after the test, it proves there is no pregnancy. Also, if the test is based on chemicals (like it usually is) there are chances of it being wrong. If the test is a scan of the abdomen and the doc found a fetus - Its confirmed.
No pregnancy test is 100% accurate. Sometimes the result is wrong, and that is shown by a subsequent test, or by later events. Depending on how much time elapses, it is also possible for a woman to become pregnant between taking the first (negative) test and the second (positive) test.
A false positive is very rare but it is possible. It is a LOT more common to get a false negaive than a false positive. The best thing for you to do, is to do another pregnancy test. If the test is still positive then you are pregnant and should see your doctor.
It is possible to get a false negative but not a false positive. If it is a cheap test or if it is too early in the pregnancy, the at home test may not recognize the hormone that the test is checking for. However, if you get a positive it means the hormone is present and that she is pregnant.