Novel dose-finding designs, using estimation to assign the best estimated maximum- tolerated-dose (MTD) at each point in the experiment, most commonly via Bayesian techniques, have recently entered large-scale implementation in Phase I cancer clinical trials. We examine the small-sample behavior of these "Bayesian Phase I" (BP1) designs, and also of non-Bayesian designs sharing the same main "long-memory" traits (hereafter: LMP1s). For all LMP1s examined, the number of cohorts treated at the true MTD (denoted here as n*) was highly variable between numerical runs drawn from the same toxicity-threshold distribution, especially when compared with "up-and-down" (U&D) short-memory designs. Further investigation using the same set of thresholds in permuted order, produced a nearly-identical magnitude of variability in n*. Therefore, this LMP1 behavior is driven by a strong sensitivity to the order in which toxicity thresholds appear in the experiment. We suggest that the sensitivity is related to LMP1's tendency to "settle" early on a specific dose level - a tendency caused by the repeated likelihood-based "winner-takes-all" dose assignment rule, which grants the early cohorts a disproportionately large influence upon experimental trajectories. Presently, U&D designs offer a simpler and more stable alternative, with roughly equivalent MTD estimation performance. A promising direction for combining the two approaches is briefly discussed (note: the '3+3' protocol is not a U&D design).