C# switch variable initialization: Why does this code NOT cause a compiler error or a runtime error?

...
case 1:
   string x = "SomeString";
   ...
   break;
case 2:
   x = "SomeOtherString";
   ...
   break;
...

Is there something that I am not understanding about the switch statement in C#? Why would this not be an error when case 2 is used?
Edit: This code works and doesn't throw an error.