Treewidth is a graph parameter that plays a fundamental role in several structural and algorithmic results. We study the problem of decomposing a given graph G into node-disjoint subgraphs, where each subgraph has sufficiently large treewidth. We prove two theorems on the tradeoff between the number of the desired subgraphs h, and the desired lower bound r on the treewidth of each subgraph. The theorems assert that, given a graph G with treewidth k, a decomposition with parameters h,r is feasible whenever hr² ≤ k/polylog(k), or h³r ≤ k/polylog(k) holds. We then show a framework for using these theorems to bypass the well-known Grid-Minor Theorem of Robertson and Seymour in some applications. In particular, this leads to substantially improved parameters in some Erdos-Posa-type results, and faster algorithms for some fixed-parameter tractable problems.