The purpose of this study is to see if adding 1 or 2 drugs to the anti-HIV therapy of patients whose HIV levels increased while taking their anti-HIV drugs can lower viral load (amount of HIV in the blood) and keep it low up to Week 24. (This study has been changed. Previously, only patients whose levels increased on their first round of anti-HIV drugs were being studied.) Anti-HIV drug treatments that contain a combination of 3 or more drugs can lower HIV levels, raise CD4 cell counts, and improve survival. Unfortunately, many patients "fail" their anti-HIV drug treatment when their HIV levels go above 500 copies/ml. Usually the next step is to switch the patient to different anti-HIV drugs. Doctors would like to see whether adding 1 or 2 different drugs to the "failed" treatment also can lower HIV levels. Adding 1 or 2 drugs might be better than switching all of the drugs since patients who take many different drugs can develop drug-resistant HIV. (This study has been changed. Previously, only patients taking protease inhibitors (PI) whose levels increased on their first round of anti-HIV drugs were being studied.)