We investigate time-dependent tunneling of wave packets through double-barrier structures with trapezoidal potential profiles. The potential profiles have been taken up as a model for structures in which the flat interfaces have been disturbed by roughened surfaces with interdiffusion of the constituent atoms. Time-dependent Schroedinger's equation, using effective mass approximation, is solved numerically. Studying the effects of the slope widths on tunneling time, lifetime and build-up time, for the particular case of the AlAs/GaAs/AlAs double-barrier structure having the barrier width of 2.3 nm and the well width of 5.0 nm, we show the slope width less than 0.6 nm does not significantly change the tunneling time. The lifetime and build-up time are shown to have different characteristics.