Current test-time adaptation (TTA) approaches aim to adapt to environments that change continuously. Yet, when the environments not only change but also recur in a correlated manner over time, such as in the case of day-night surveillance cameras, it is unclear whether the adaptability of these methods is sustained after a long run. This study aims to examine the error accumulation of TTA models when they are repeatedly exposed to previous testing environments, proposing a novel testing setting called episodic TTA. To study this phenomenon, we design a simulation of TTA process on a simple yet representative $\epsilon$-perturbed Gaussian Mixture Model Classifier and derive the theoretical findings revealing the dataset- and algorithm-dependent factors that contribute to the gradual degeneration of TTA methods through time. Our investigation has led us to propose a method, named persistent TTA (PeTTA). PeTTA senses the model divergence towards a collapsing and adjusts the adaptation strategy of TTA, striking a balance between two primary objectives: adaptation and preventing model collapse. The stability of PeTTA in the face of episodic TTA scenarios has been demonstrated through a set of comprehensive experiments on various benchmarks.