Why won't the bulbs light even when the fixture is has power?
You probably need to replace the base (where the lamp goes) in the fixture. Keep in mind that your tester does not draw much of anything at all, so it won't detect a bad connection in the base that shows up when the lamp itself is actually pulling something.

Check also that the connections from the box to the fixture are solid.

If the fixture itself is cheap, consider just replacing it completely. If you continue to have problems, the loose connection isn't in the light fixture.

You can quickly determine this by attaching a pig tail light:



.. directly to the feeders in the light fixture. If the pig tail works, the problem is the fixture (most likely the base). If the pig tail doesn't work, you may have a bad switch, loose neutral, or something else that an electrician should probably inspect.