Artificial Intelligence (AI) systems are attracting increasing interest in the medical domain due to their ability to learn complicated tasks that require human intelligence and expert knowledge. AI systems that utilize high-performance Natural Language Processing (NLP) models have achieved state-of-the-art results on a wide variety of clinical text processing benchmarks. They have even outperformed human accuracy on some tasks. However, performance evaluation of such AI systems have been limited to accuracy measures on curated and clean benchmark datasets that may not properly reflect how robustly these systems can operate in real-world situations. In order to address this challenge, we introduce and implement a wide variety of perturbation methods that simulate different types of noise and variability in clinical text data. While noisy samples produced by these perturbation methods can often be understood by humans, they may cause AI systems to make erroneous decisions. Conducting extensive experiments on several clinical text processing tasks, we evaluated the robustness of high-performance NLP models against various types of character-level and word-level noise. The results revealed that the NLP models performance degrades when the input contains small amounts of noise. This study is a significant step towards exposing vulnerabilities of AI models utilized in clinical text processing systems. The proposed perturbation methods can be used in performance evaluation tests to assess how robustly clinical NLP models can operate on noisy data, in real-world settings.