A hydraulic system for a mobile machine having an implement is disclosed. The hydraulic system may have a hydraulic actuator configured to move the implement, a storage device configured to store pressurized fluid, and a valve operable to fluidly communicate the storage device with the hydraulic actuator. The hydraulic system may also have a sensor associated with the mobile machine to generate a signal indicative of a speed of the mobile machine, and a controller in communication with the valve and the sensor. The controller may be configured to compare the speed of the mobile machine to a setpoint and to determine an amount of time elapsed while the speed of the mobile machine exceeds the setpoint. The controller may further be configured to selectively operate the valve to fluidly communicate the storage device with the actuator based on the elapsed amount of time and the speed to cushion movement of the implement.