Developers not writing tests
The largest problem your test suite will have is developers that won't write tests. If there is a test suite and the test is run upon committing new code, then the tests will run when these developers commit code to the repository. However, the new code will not be covered by tests and there is uncertainty as to what will happen if the commit breaks the existing tests. Will the developer fix these tests or leave the test suite red?
The impact of not writing tests will start to devalue the whole test suite as the coverage goes down. If some developers stop updating the test suites, then others will follow, and soon you will have a test suite without any active development. Then, the team will not get full benefits, such as enabling refactoring or feeling safe about new features.
I have experienced development teams disabling a whole test suite because a test was failing. They claimed it was because they needed a green build in order to deploy to production...
