Researchers analyzing longitudinal data often want to find out whether the process they study is characterized by (1) short-term state variability, (2) long-term trait change, or (3) a combination of state variability and trait change. Classical latent state-trait (LST) models are designed to measure reversible state variability around a fixed set-point or trait, whereas latent growth curve (LGC) models focus on long-lasting and often irreversible trait changes. In the present article, we contrast LST and LGC models from the perspective of measurement invariance testing. We show that establishing a pure state-variability process requires (1) the inclusion of a mean structure and (2) establishing strong factorial invariance in LST analyses. Analytical derivations and simulations demonstrate that LST models with noninvariant parameters can mask the fact that a trait-change or hybrid process has generated the data. Furthermore, the inappropriate application of LST models to trait change or hybrid data can lead to bias in the estimates of consistency and occasion specificity, which are typically of key interest in LST analyses. Four tips for the proper application of LST models are provided.