It seems your drug gives absorbance independent of the cellular activity. Check your medium+drug absorbance (without cells and use same drug concentration)
When you add a higher drug concentration, are you simply adding more volume of your chemical stock, or have you normalized the volume added (ie, made a more concentrated chemical stock, so you add the same chemical stock volume for all concentrations)? If the former - and you don't have controls built in to serve as an appropriate reference control - this could be causing the discrepancy. Make sure you have the appropriate negative controls (as Tuncay suggested) to sort out the issue.
does your drug has color? try to look at your cells before staining to see if they are really living or the increase in absorbance is just due to the color effect. keep in your mind that some drugs have optimum concentrations where the cell take the highest amount of the drug to the inside. above and below this concentration the uptake is less so the viability is high which may explain what happened with you.
Does your drug has high density or viscous or like suspension it might absorb some light. And i am also agree with Wamidh H Talib as a drug give best result at some concentration above that it may not work or behave differently too