TDD question about starting with a failing test

I'm fairly new to TDD. I've been using it for several months and it's been pretty great so far.

Now I have a question about what it means to "start with a failing test." As I've done TDD, I find myself frequently running into the following situation:

• I write a failing test for a simple test case (say, test case 1)

• I then write code that passes test case 1

• I'll write another test for a different test case (say, test case 2)

• The code that was able to pass test case 1 is also able to pass test case 2

So in other words, my test for test case 2 began as passing instead of failing. Is that a problem? Does that violate the "begin with a failing test" rule?

If it is a problem, does that mean I should get rid of the test for the second test case? Should I only create new tests for test cases that cause failures?