In TFS Source Control Explorer, in some projects the "New Folder" and "Add Files" icons are visibles, and for other projects they are not. I'm a contributor in both, and I can check in solutions in both, and not an admin for either.
What could possibly be making the icons go "dim". Is it security? Or some other combination of rules?
First of all, make sure that you have that folder mapped in your current workspace. If it isn't mapped, you wont' be able to add files. Based on you saying that you can't even click on the Add Files button, that makes me think this is the issue.
If that's not the case, and you need to drill down into security, I highly recommend the free Team Foundation Sidekicks from Attrice, which you can find at http://www.attrice.info/
Look at their "Permissions Sidekick" within the tool.