When people say "we can't test that", they usually mean "... with a reasonable effort". They say "we can't test that because it's using a database" or "we can't test the layout of the UI" or "to test this, we need information which is buried in private fields of that class".
And they are always wrong. You can test everything. Usually with a reasonable effort. But often, you need to take a step back and do the unusual. Some examples.
So your app is pumping lots of data into a database. You can't test the database. You'd need to scrap it for every test run and build it from scratch which would take hours or at least ages. Okay. Don't test the database. Test how you use it. You're not looking for bugs in the database, you're looking for bugs in your code. Saying "but some bugs might get away" is just a lame excuse.
Here is what you need to do: Identify independent objects (which need no other objects stored in the database). Write tests for those. Put the test data for them in an in-memory database. HSQLDB and Derby are your friends. If you must, use your production database but make the schema configurable. Scrap the tables before the test and load them from clean template tables.
So you need some really spiffy SQL extensions? Put them in an isolated place and test them without everything else against the real database. You need to test that searching a huge amount of data works? Put that data in a static test database. Switch database connections during the tests. Can't? Make that damn connection provider configurable at runtime! Can't? Sure you can. If everything else fails, get the source with JAD, compile that into an independent jar and force that as the first thing into the classpath when you run your tests. Use a custom classloader if you must.
While this is not perfect, it will allow you to learn how to test. How to test your work. Testing is always different just like every program is different. Allow yourself to make mistakes and to learn from them. Tackle the harder problems after the easier ones. Make the tests help you learn.
So you have this very complex user interface. Which you can't test. Let alone starting the app takes ten minutes and the UI changes all the time and ... Okay. Stop the whining. Your program is running on a computer and for same inputs, a computer should return the same outputs, right? Or did you just build a big random number generator? Something to challenge the Infinite Improbability Drive? No? Then you can test it. Follow me.
First, cut the code that does something from the code that connects said code to the UI. As a first simple step, we'll just assume that pressing a button will actually invoke your method. If this fails for some reason, that reason can't be very hard to find, so we can safely ignore these simple bugs for now.
After this change, you have the code that does stuff at the scruff. Now, you can write tests for it. Reduce entanglement. Keep separate issues separate. A friend of mine builds all his code around a central event service. Service providers register themselves and other parts of the code send events to do stuff. It costs a bit performance but it makes testing as easy as overwriting an existing service provider with a mock up.
Your software needs an insanely complex remote server? How about replacing this with a small proxy that always returns the same answers? Or at least fakes something that looks close enough to a real answer to make your code work (or fail when you're testing the error handling).
And if you need data that some stubborn object won't reveal, use the source, Luke (download the source and edit the offender to make the field public, remove "final" from all files, add a getter or make it protected and extend the class in the tests). If everything else fails, turn to java.lang.reflect.Field.setAccessible(true).
If you're using C/C++, always invoke methods via a trampoline: Put a pointer somewhere which contains the function to call and always use that pointer instead of the real function. Use header files and macros so no human can tell the difference. In your tests, bend those pointers. The Amiga did it in 1985. #ifdef is your friend.
If you're using some other language, put the test code in comments and have a self-written preprocessor create two versions that you can compile and run.
If all else fails, switch to Python.