We use TestNG and Selenium WebDriver to test our web application.
Now our problem is that we often have several tests that need to run in a certain order, e.g.:
You are mixing "functionality" and "test". Separating them will solve your problem.
For example, create a helper class/method that executes the steps to log in, then call that class/method in your Login test and all other tests that require the user to be logged in.
Your other tests do not actually need to rely on your Login "Test", just the login class/method.
If later back-end modifications introduce a bug in the login process, all of the tests which rely on the Login helper class/method will still fail as expected.
Update:
Turns out this already has a name, the Page Object pattern. Here is a page with Java examples of using this pattern:
http://code.google.com/p/selenium/wiki/PageObjects