Motivated by the recently observed intriguing mode splittings in a magnetic field with inelastic neutron scattering in the spin ladder compound (C5H12N)(2) CuBr4 (BPCB), we investigate the nature of the spin ladder excitations using a density matrix renormalization group and analytical arguments. Starting from the fully frustrated ladder, for which we derive the low-energy spectrum, we show that bound states are generically present close to k = 0 in the dynamical structure factor of spin ladders above H-c1, and that they are characterized by a field-independent binding energy and an intensity that grows with H - H-c1. These predictions are shown to explain quantitatively the split modes observed in BPCB.