We study families of Boolean circuits with the property that the number of gates at distance t fanning into or out of any given gate in a circuit is bounded above by a fixed polynomial in t . This polynomial bound, say O(t^k) , is independent of the input length n or size of circuits C_n in the family. We prove that such circuits require non-linear size \Omega(n^{1+1/k}/\log n) to compute several basic functions, including sorting, finite-field arithmetic, and certain linear transformations. These results imply a ``size-vicinity tradeoff'' for circuits. Our proof develops a ``separator theorem'' in the style of Lipton and Tarjan [1979] for a new class of graphs, and our proof methods seem to have independent interest and applications.