Lets say you are testing a application. You have written a set of test cases.
Now you run one cycle of testing. You find few bugs and report them to the development team. Development team fixes the bugs and reverts to you with the updated code. You again execute the same set of test cases. This time you find that few of the bug were still not fixed and you report that back to the development team. They work of it and send an update to you. Once again, you execute the same set of test cases and don't find any bugs.
Now in a new release some changes were made in the application. You run the same set of test cases and they all pass. But, what you miss here is the new bugs that may have introduced when the fix and new changes were applied. The old set of test cases are incapable of identifying these new bugs.
This is called Pesticide Paradox. To avoid this you need to update your test cases with each cycle and add new cases to the old set.