Yes, what you are referring to is called cascading. If you consider the output of the first stage, it has a 12db/oct rolloff. Now if you put that into the second stage it will operate on the signal in complete isolation of the first stage so if it has another 12dB/oct rolloff at the same corner frequency then your final output will have 24dB/oct rolloff. In other words you just if you plot the frequency response of the two filters and add them together you'd get the overall response of the system.
Generally you think of filters as attenuating the signal only, but fiddling with the Q or using the peaking filter for example can give you boost at certain frequencies. When you consider the filter responses being added together then you also have to realize that the gains can add together and cause clipping.
I'm not familiar with the way web audio goes about it's filter design so I would suggest either poking around in the code (if it's available) or measuring your filter's frequency response.