What OS are you using, Mac or windows?
Have you checked the permissions on the drive/folder you are copying or moving the images to?
If you already have the images on your drive and you are selecting ADD in the import dialog check the permissions on the drive/folder the image are on.
What is the exact error message?
Is the import failing for all the files in the group or just some of them?
Thanks guys. The problem was not with LR but with my desktop not having enough storage space. Something I should have checked earlier.