We present a theory for placing a lower bound on the thickness of the oceanic magnetized layer using magnetic anomaly observations and estimates of the intensity of magnetization; the theory makes only a minimum number of assumptions regarding the spatial distribution of the magnetization. The principle of the method is based upon the fact that thin layers imply high magnetizations. We show how to calculate the source distribution that has minimum intensity yet fits the data and is confined to a given thickness layer; because the minimum intensity must be a monotonically decreasing function of layer thickness, it follows that an upper bound on the intensity allows us to obtain a lower limit on the thickness. The practical caluclations are performed by using linear programing. The method is applied to two sets of near-bottom magnetic profiles, one on the Gal¿pagos Spreading Center at 86¿W and the other sets on the Pacific-Antarctic Ridge at 51¿S. In the first area we conclude that the magnetic layer must be at least 450 m thick, and in the other a crossing of the Jaramillo event indicates that the magnetic layer is probably more than 1000 m in thickness.