Nonlinear wave motion is studied in a symmetric, continuously stratified, smoothed three-layer fluid in the framework of the fully nonlinear Euler equations under the Boussinesq approximation. The weakly nonlinear limit is discussed in which the governing equations can be reduced to the fully integrable modified Korteweg-de Vries equation. For some choices of the layer thicknesses the cubic nonlinear term is positive and the modified Korteweg-de Vries equation has soliton and breather solutions. Using such a stratification, the Euler equations are solved numerically using a sign-variable, initial disturbance. Breathers were generated for several forms of the initial disturbance. The breathers have moderate amplitude and to a good approximation can be described by the modified Korteweg-de Vries equation. As far as we know this is the first presentation of a breather in numerical simulations using the full nonlinear Euler equations for a stratified fluid.