I have this problem sometimes with Mass Effect 1 (2007, classic edition). Don't know why it happens, but I figured out when it does there is a zombie ActivationUI.exe and Mass Effect.exe running in the background. I end those tasks and try again until it works. Sometimes it just works. Other times it takes a few tries.
@Steelydan73 You right click the .exe file in the folder and choose Properties. Click Compatibility tab and check the option towards bottom to "run this program as an administrator". Repeat this for each one in the folder.