An airbag assembly has an airbag, a gas emitting inflator, and an airbag thickness limiting structure. The airbag thickness limiting structure limits a first body region engaging portion to having a lateral thickness within a first lateral thickness range when the gas emitting inflator delivers a first amount of inflation gas into the interior space of the airbag and allows the first body region engaging portion of the airbag to have a lateral thickness within a second lateral thickness range when the gas emitting inflator delivers a second amount of inflation gas greater than the first amount into the interior space of the airbag. A second body region engaging portion has a substantially fixed maximum attainable lateral thickness. Maximum attainable lateral thickness of the first body region engaging portion of the airbag can be dependent upon whether or not inflation of the airbag is implemented using pre-crash sensing technology.