We investigate topologically biased failure in scale-free networks with a degree distribution P(k)proportional to k(-gamma). The probability p that an edge remains intact is assumed to depend on the degree k of adjacent nodes i and j through p(ij)proportional to(k(i)k(j))(-alpha). By varying the exponent alpha, we interpolate between random (alpha=0) and systematic failure. For alpha > 0 (< 0) the most (least) connected nodes are depreciated first. This topological bias introduces a characteristic scale in P(k) of the depreciated network, marking a crossover between two distinct power laws. The critical percolation threshold, at which global connectivity is lost, depends both on gamma and on alpha. As a consequence, network robustness or fragility can be controlled through fine-tuning of the topological bias in the failure process.