I have a Visual Studio 2013 solution that is under source control (TFS, Visual Studio Online). It is mapped to a local workspace, I have checked out some files (e.g. web.config,
Extensions might be the problem. Or you could be opening some readonly files. It's worth to mention that some of the extensions can turn any file you create in to a readonly file, and this would be the same result that you're facing.
So first, disable all extensions, and check for the little icon near the files's name (in the tab) that indicates a readonly file.