When is acceleration not constant?
Acceleration is not constant if the net force is not constant.
Imagine a rocket propelled car. It has 3 rockets on the back. All the rockets ignite and forward acceleration is constant at 3g's. Then one of the engines has a malfunction and quits. The acceleration changes to 2 g's. The acceleration was not constant because the force was not constant.
I hope this help,
P.S. In cases where acceleration is constant at some non-zero value, velocity will be changing.