A speaker works by shoving air forward...active noise cancellation works by sucking in the diaphragm at the same moment the 'force' of outward air to 'suck in' the energy. Microphones and knowing the speed that sound travels is what allows the noise cancellation circuitry to know exactly when to provide that 'hollow' space to cancel the noise.

No noise cancelling headphone cancels deeper frequencies as well as higher ones...the bigger the diaphragm the lower the frequency you can 'catch'. Most noise cancelling starts with passive noise cancelling - which is a fancy word for 'plugging your ears' - ala foam/vinyl/leather to block the sound from getting in. In smaller air buds you're automatically getting passive noise cancelling just by shoving them in your ears. Also your ears can only 'hear' the most powerful push...your eardrum is a diaphragm too. But it can only 'hear' the loudest thing at any given time. Many active headphones just produce a white noise 'hiss' which is amazingly effective at just drowning out the randomness of noise...and your brain just loves it as it only takes a few minutes for your brain to stop hearing that hiss/hizz noise.

Edit: Consider stereo systems...the bigger the speaker the deeper the bass. No one has 'tweeters' thumping the bass. And in ear phones have tricks to increase the bass - but the bigger the driver the better the bass sound you can get.