One of the projects I worked on was heavily unit-tested; we had over 1000 unit tests for 20 or so classes. There was slightly more test code than production code. The unit tests caught innumerable errors introduced during refactoring operations; they definitely made it easy and safe to make changes, extend functionality etc. The released code had a very low bug rate.
To encourage ourselves to write the unit tests, we specifically chose to keep them 'quick and dirty' - we would bash out a test as we produced the project code, and the tests were boring and 'not real code', so as soon as we wrote one that exercised the functionality of the production code, we were done, and moved on. The only criteria for the test code was that it fully exercised the API of the production code.
What we learnt the hard way is that this approach does not scale. As the code evolved, we saw a need to change the communication pattern between our objects, and suddenly I had 600 failing unit tests! Fixing this took me several days. This level of test breakage happened two or three times with further major architecture refactorings. In each case I don't believe we could reasonably have foreseen the code evolution that was required beforehand.
The moral of the story for me was this: unit-testing code needs to be just as clean as production code. You simply can't get away with cuttting and pasting in unit tests. You need to apply sensible refactoring, and decouple your tests from the production code where possible by using proxy objects.
Of course all of this adds some complexity and cost to your unit tests (and can introduce bugs to your tests!), so it's a fine balance. But I do believe that the concept of 'unit tests', taken in isolation, is not the clear and unambiguous win it's often made out to be. My experience is that unit tests, like everything else in programming, require care, and are not a methodology that can be applied blindly. It's therefore surprising to me that I've not seen more discussion of this topic on forums like this one and in the literature.