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We propose a bootstrap procedure for data that may exhibit cluster-dependence in two or more dimensions. The asymptotic distribution of the sample mean or other statistics may be non-Gaussian if observations are dependent but uncorrelated within clusters. We show that there exists no procedure for estimating the limiting distribution of the sample mean under two-way clustering that achieves uniform consistency. However, we propose bootstrap procedures that achieve adaptivity with respect to different uniformity criteria. Important cases and extensions discussed in the paper include regression inference, U- and V-statistics, subgraph counts for network data, and non-exhaustive samples of matched data.