It depends, really.
In a language like C++, you can get away with this because the compiler will let you know right away if something would break. However, other less-picky languages will allow you to refer to variables which don't exist, and the worst that happens is a slap on the wrist in the form of an exception being thrown for a null reference.
I was working on a flex project once where the codebase was a real mess, and we decided to go through the code and beautify it a bit to meet the Adobe AS3 coding standards. Since I was new to the project, I didn't realize that the variable names in some classes actually referred to persistent objects which hibernate (running the java webapp for the backend server) was using to create mappings. So renaming these variables caused the entire flex frontend to misbehave, even when we did it with the "correct" refactoring tools in our IDE.
But really, I'd say to check your OCD at the door and make your changes a little at a time. Any time you change dozens of files in a large project, you risk destabilizing it, and in this case, the benefit derived from such a risk doesn't pay off.