The low back is more suited for stability than mobility. When someone has tight hips or a tight thoracic spine, range of motion at the lumbar spine tends to increase as a compensation. Over time, this makes the lumbar spine more susceptible to injury, and whether it's picking something up, or simply getting up after sitting on the floor as tennisenthusiast mentioned, it is often an otherwise harmless movement that is the last straw and causes pain or injury.
TheCheese is right in that you need trunk stability to stabilize the lumbar spine during movement, but you also need good hip and thoracic spine mobility to avoid compensation at the lumber spine. Without hip and t-spine mobility, the strongest core in the world won't make that much difference. The forces generated by sports and daily activity will win.