Much research on software testing makes an implicit assumption that test failures are deterministic such that they always witness the presence of the same defects. However, this assumption is not always true because some test failures are due to so-called flaky tests, i.e., tests with non-deterministic outcomes. Unfortunately, flaky tests have major implications for testing and test-dependent activities such as mutation testing and automated program repair. To deal with this issue, we introduce a test flakiness assessment and experimentation platform, called FlakiMe, that supports the seeding of a (controllable) degree of flakiness into the behaviour of a given test suite. Thereby, FlakiMe equips researchers with ways to investigate the impact of test flakiness on their techniques under laboratory-controlled conditions. We use FlakiME to report results and insights from case studies that assesses the impact of flakiness on mutation testing and program repair. These results indicate that a 5% of flakiness failures is enough to affect the mutation score, but the effect size is modest (2% - 4% ), while it completely annihilates the ability of program repair to patch 50% of the subject programs. We also observe that flakiness has case-specific effects, which mainly disrupts the repair of bugs that are covered by many tests. Moreover, we find that a minimal amount of user feedback is sufficient for alleviating the effects of flakiness.