Are small speakers inherently limited to higher frequencies? I am hoping to build a subwoofer using multiple smaller speakers (165mm) instead of a single larger speaker (380mm).
My theory is that the displaced air volume is what matters, not the individual speaker's diameter.
The surface area of a 380mm speaker is $\approx900cm^2$.
The surface area of a 165mm speaker is $\approx140cm^2$.
Presuming the speakers are capable of being driven to the same $X_\max$, six or seven 165mm speakers should be roughly equivalent to a 380mm speaker.
Taken to an extreme, an array of 100 $9cm^2$ speakers should be equivalent to a single $900cm^2$ speaker.
Is this correct, or do smaller speakers have some other limiting factor that limits their low-frequency reproduction?
 A: It can be done, but there's some trade offs.  Larger speakers are better at moving longer wavelength (low frequency) waves.  When you try to combine a bunch of small surfaces in different locations to recreate a single wave you end up with a some random interference where the wave is stronger or weaker (in 3d-space) (see phased-array antenna for some examples).  Whereas with one large speaker you get a more uniform reproduction of the wave in all directions.
In addition the size of the speaker can also aid in the how well it transmits the wave.  Larger speakers better match the frequency of the low sounds so they are broadcast better and with more force than it would be on 3 smaller speakers with the equivalent surface area.

Edit:  Typically woofers are built different than midrange or tweeters because they need to move a lot more air to create the powerful low frequency sounds you bounce to at the clubs.  Midrange/tweeter speakers are not meant to be driven hard at low frequencies so you have to be careful with the amplification when driving these speakers outside of their optimized frequency range.

The design is optimized for low frequency response and allows the sound to resonate better (more efficiently).  Likewise they don't handle the higher frequencies as well so this is why a filter called a cross over network is used to send the high notes to the tweeter and low to the bass.

Part of the design that is important is getting the box to also resonate with the bass because this will help project a stronger and cleaner sound and keep the backside of the low frequency wave from canceling the front side.  This need to isolate sound waves between the front and back of an open-back driver does not exist at all frequencies. It occurs only for lower frequencies whose wavelength is relatively large when compared to the diameter of the cone. Because the diaphragm becomes directional as the wavelengths become shorter and so the sound will naturally not mix between the front and back even when the driver is in open air. This is why a driver in open air lacks bass or sounds "thin". It produces mid and high frequencies without the rear sound waves canceling the front ones. But the low frequencies are diminished or canceled.

Hopefully this helps you understand the differences in performance due to design.  You can drive small speakers fine at low frequencies, look at headphones for example.  However if you want to bounce the house with a powerful bass, you need a big diaphragm.  And if you want to go cheap with a bunch of smaller ones, you have to be careful not to over drive them at low frequencies and you'll have to play with the box some to get the sound right.  In the end the results will probably be better with one bass speaker instead of trying to wire in and adjust 9 others.
And naturally there is a limit to how well you can build something so that all their sounds combine to equal the surface area of one good speaker.  As in your extreme example the 100 speakers are going to have a very weak low frequency sound and they are going to be spread out over a very big area.  Getting that sound to combine right and not cancel each other out is probably impossible.  You'll have 100 speakers all operating at different phases mixing and canceling each other around the room.

A: There is a very sound mathematical / physical explanation why more small speakers will not work very well (perhaps only with very low power, just like in a headphone).
In order to feed a constant sound power in the air, the excursion of a speaker with a certain surface must decrease its displacement with the square of the frequency. So, low frequencies require very large excursions.
Above the resonance frequency of the speaker the amplitude of the cone is mainly damped by the airforce. This means that the electrical power is converted into sound power or air pressure. Above a certain frequency the mass of the cone will play its role to decrease the amplitude; hence the speaker will have its upper frequency limit.
Below the resonance frequency of the speaker, the movement of the cone is mainly determined by the stiffness added with the air pressure built up in a closed box (so not by the movements of air). The amplitude of the cone will hence be constant at a certain voltage (and supplied power) independent of the frequency. This implies that the air sound pressure diminishes with the square of the frequency below the resonance frequency of the speaker(s).
So, a bunch of small speakers with an eigenfrequency of e.g. 70 Hz will not solve this problem and will never produce e.g. 30 Hz with sufficient sound pressure.
A solution could be, when you use small speakers with very low stiffness and sufficient volume in the box  (thus with a low eigenfrequency!); of course either the excursion distance must then be large enough or the supplied power not too big; otherwise the cone will clip to the end of its free moving space.
See for the extended mathematical explanation:
https://rmsacoustics.nl/papers/whitepapersoundgeneration.pdf
An example of a relative small 8" speaker with low eigenfrequency is the KEF SP1039 with 25 Hz. The modern 8" Visaton WS20 scores worse with 47 Hz. 
When I read the above mentioned article I finally understood why the bass in my KEF 104 was degraded so much, when I replaced a dead SP1039 with a the well-fitting WS20. So, finally I bought a used SP1039 to restore the good basses of my KEF 104.
