We investigate physical models which possess simultaneous ordinary and type A N -fold supersymmetries, which we call type A (N , 1)-fold supersymmetry. Inequivalent type A (N , 1)-fold supersymmetric models with real-valued potentials are completely classified. Among them, we find that a trigonometric Rosen-Morse type and its elliptic version are of physical interest. We investigate various aspects of these models, namely, dynamical breaking and interrelation between ordinary and N -fold supersymmetries, shape invariance, quasi-solvability, and an associated algebra which is composed of one bosonic and four fermionic operators and dubbed type A (N , 1)-fold superalgebra. As realistic physical applications, we demonstrate how these systems can be embedded into Pauli and Dirac equations in external electromagnetic fields.