We study N=2 supersymmetric Born-Infeld-Higgs theory in 3 dimensions and derive Bogomol'nyi relations in its bosonic sector. A peculiar coupling between the Higgs and the gauge field (with dynamics determined by the Born-Infeld action) is forced by supersymmetry. The resulting equations coincide with those arising in the Maxwell-Higgs model. Concerning Bogomol'nyi bounds for the vortex energy, they are derived from the N=2 supersymmetry algebra.Facultad de Ciencias Exactas