We solve a class of doubly reflected BSDEs arising naturally in the study of Convertible Bonds in finance. A specificity of this class of doubly reflected BSDEs is that the barriers are not separated, which makes the problem non standard in the doubly reflected BSDEs literature. In a generic Markovian set-up, we establish the connection between the solution of the doubly reflected BSDE and the viscosity solution of a related system of coupled double obstacle problems.