If you have a test project, you'll get tired of keeping your configs in sync. Depending how you're project is setup, sometimes you can refer some sections to a centralized config. However most of the time I prefer to just use a post build event to copy the file from one project to another.
copy $(SolutionDir)MyWebProject\Web-ConnectionStrings.config $(ProjectDir)MyProject.Tests.ConnectionStrings.config
echo ^<!--copied via post-build event in $(ProjectName)--^> >>$(ProjectDir)MyProject.Tests.ConnectionStrings.config
The first line copies the file. But I wanted myself or other developers to know that the file is being overwritten, so I added the second line which appends a message to the end of the file.