Pulse train instabilities have often given rise to confusion and misinterpretation in ultrafast pulse characterization measurements. Most prominently known as the coherent artifact, a partially mode-locked laser with a non-periodic waveform may still produce an autocorrelation that has often been misinterpreted as indication of a coherent pulse train. Some modern pulse characterization methods easily miss the presence of a coherent artifact, too. Here, we address the particularly difficult situation of a pulse train with chirp-only instability. This instability is shown to be virtually invisible to autocorrelation measurements, but can be detected with frequency-resolved optical gating, spectral phase interferometry for direct electric-field reconstruction, and dispersion scan. Our findings clearly show that great care is necessary to rule out a chirp instability in lasers with an unclear mode-locking mechanism and in compression experiments in the single-cycle regime. Among all dynamical pulse train instabilities analyzed so far, this instability appears to be the best-hidden incoherence and is most difficult to detect.