My understanding is that a larger cabinet will support a lower frequency response in the bass. The shape of that frequency response is also shaped by the design of the cabinet, i.e., whether it be sealed box or bass reflex (let's leave out the other designs for the sake of keeping this simple).
As many of you already know, the sealed box design will make a slower slope in the bass response (although not as deep) than the bass reflex, which goes lower in frequency, yet at a higher slope than the sealed design. Given this, bass response is altered by the physical size of the cabinet. A smaller cabinet simply cannot load the woofer for lower frequencies. Of course, these factors much match the T/S parameters of the driver to begin with to make a reasonable bass response in the first place. A bigger cabinet allows for lower and possibly flatter response in general than trying to make a small woofer and cabinet do what it cannot. A popular trick used for smaller cabinet/woofer sizes is to make use of a bump usually at 100hz. It gives the likeness of more bass, a thump, if you will, but it is not ideal for my tastes.
This is the case for passive speakers, but active speakers can use EQ in the signal to try and offset the limitations of the cabinet and driver. In the end, physics will rule up to the limits of the final product.