I think I can say something about how a blown driver sounds, and I believe your driver is blown on one side.
I have had pairs of defective AKG, Sennheiser, and Beyerdynamic before, and they all suffered the same defects. First it was my first K271 Studio, it was rattling in one driver at very low bass. Next it was Sennheiser PX100, and the latest was my Beyerdynamic DT770 Pro. They all had rattling in one driver at very low bass notes. Mind you, I didn't even play loud. I just connect one of my synthesizers and play a very low bass note at a moderate volume, and they rattled. They sounded like their driver was torn and vibrated at a low frequency. I felt like I had really bad luck with all my headphone purchases.
You usually hear them work normally without any artifacts until you go down low enough in frequency, until, I think, the frequency resonates with the tear in the driver, thus creating flapping sound at a specific low frequency range.
Long story short, I sent them back and got them replaced. All these models are still with me. After the replacement, they no longer have rattles at any frequency.
Later on, I bought me a pair of AKG K702, and it was among the very few that I didn't have to replace (another one was Grado SR80, which I have sold later). And its bass? Sure, it's not in the same ballpark as my DT770 Pro, but I think it's just enough -- albeit just a little too dry sometimes.
Do I hear low frequencies on my K702? Well, you bet, just not very prominent, and the closer to 20 Hz it is, the more I feel rather than hear it. At around 20 Hz, I think I don't hear it anymore (just some very faint feeling of it), but that doesn't bother me much. I'm used to the sound of small near field monitors (8" drivers or less), and that kind of extremely low sub-bass is not reproduced by most of these anyway.
To the OP, I think it's probably not a very good idea to push the volume "close to the max" to try to hear the bass. Even though you might not hear it (either because the headphones cannot reproduce it well enough or because you simply can't hear it), the amp might have already reached its maximum potential to provide enough power to produce the bass, and is driven into clipping. Just try listening at your normal listening volume, don't push it, and if you can't hear it, then it simply doesn't have enough bass for your need.
Also, I think it's not a very fair comparison to compare the amount of bass on your Bose with the K702. Among the recording/mixing studio favorites like the AKG K240/271/70x, the Beyerdynamic DT100/150/250/770/880/990, the Sennheiser HD280/600/650, the Sony MDR-V6/7506/7509, the Audio Technica ATH-M50, the various Ultrasones, and a few others, the Beyerdynamic DT770 Pro and the Audio Technica ATH-M50 are probably among the bassiest of them all, yet from the frequency response graph from Headphone.com, these two still are less bassy than the two Bose models they have the graphs for. I think that is saying something about the Bose sound signature.
