Memory-based self-improving agents--those that learn from an online stream of tasks and improve over time by maintaining a textual memory bank--have shown great promise in recent literature. However, the reliability aspects of these methods have been critically overlooked. In this work, we conduct a comprehensive re-evaluation of two memory-based methods, broadening the scope of evaluation along two axes: (1) including multiple self-improving runs to quantify variance, and (2) shuffling the tasks to investigate the effect of task order. Through these experiments, we make two observations that expose the fragility of current methods: First, agent evaluation is inherently noisy in complex environments and on multi-step tasks, and stacking a self-improving loop on top can further amplify this noise. Second, the agent's improvement is highly dependent on task order. Prior works often adopt default orderings that impose an implicit curriculum, acting as a hidden prerequisite for success. To better understand this fragility, we manually examine the agents' memory and hypothesize that task and environment underspecification contribute to this fragility. We validate this hypothesis by incorporating information that enables better specification, such as detailed rubrics and environment feedback, into the memory construction process. While this added information partially closes the performance degradation in previous experiments, significant gaps still remain, suggesting that other uncharacterized factors contribute to this fragility. Looking ahead, our work advocates for more rigorous evaluation protocols for self-improving agents by reporting results across multiple runs and stress-testing them under challenging conditions. Moreover, our findings on underspecification call for systems and interfaces that enable effective human oversight, preventing agents from failing in unforeseeable ways.