Is it possible to have pregnancy symptoms when the test is negative?

Absolutely. Pregnancy symptoms are non-specific. This means something else could be going on other than pregnancy. Fatigue, breast tenderness, "morning sickness" could all be due a variety of other causes, most of which will resolve with time. Pregnancy tests are also not perfect. If symptoms persist see your OB for a pregnancy blood test.
It Depends. It depends on if you took an at home pregnancy test or a test from your doctor. Both tests work by detecting a hormone that is only made when you are pregnant. Home tests use urine & there may not have been enough hormone to detect at the time you took it. Blood tests can be more accurate. See your doctor about the symptoms & to get a test.