In multiple-baseline experiments, staggered adoption designs, and stepped-wedge trials, every unit switches once from baseline to treatment at a randomized onset time and remains treated afterward. The treatment onsets in these designs are absorbing. Since such experiments often involve few heterogeneous units, randomization tests of the sharp null of no effect on any unit-period cell are common. Rejecting this null, however, implies that some effect exists somewhere, not that it is large, positive for most unit-periods, or persistent. To improve on this limitation, we study what can be established via randomization tests for bounded and quantile nulls on cell effects in absorbing onset designs. Under the assumption that a cell's potential outcome depends on its current treatment status and not on when treatment began, we establish finite-sample validity for both tests. We then develop the power theory for the bounded-null tests with a fixed number of units $N$ and a growing series length $T$ admitting $K_T$ onsets. The $p$-value against a fixed alternative decays as $K_T^{-N}$ up to a factor polylogarithmic in $T$, with lower bounds of the same order, so the onset window, not the series length, determines the power. Theory-implied guidelines for experiment planning are also provided.

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