I suppose you can consider this as the "follow-up" to my original essay on why Quantum Mechanics is so difficult. But really, this is tackling a different issue, and it is something that some time even physicists have mentioned, often to convey a message to the general public. Unfortunately, more often than not, the accuracy of the message gets distorted.
The claim that "no one understands quantum mechanics" is often attributed to Richard Feyman, who said that to illustrate the perceived "randomness" that is at the heart of quantum mechanics and the Copenhagen Interpretation of QM. The unfortunate consequence of this phrase is that we now have people using it to claim that we know NOTHING about QM, and that no one understands it.
Without even going into what QM is, let's consider the following first and foremost: we have used QM to produce a zoo of devices and techniques ranging from your modern electronics to medical procedure such as MRI and PET scans, etc. Already one can question whether this is a symptom of something that no one understands? When was the last time you place your life and the lives of your loved ones in something that NO ONE understands? That is what you do when you fly in an airplane or drive in a car that nowadays use modern electronics. All of these depend on QM for their operations!
The issue here is what is meant by the word "understand". In physics, and among physicists, we usually consider something to be "fully understood" when it has reached a universal consensus that this is the most valid description of a phenomenon. We say that we understand Newton's Laws because it is well-tested and we know that it definitely work within a certainly range of condition. No one would question their use when building a house, for example. The same can be said about superconductivity before 1986. The BCS theory was so successful that it was of general consensus that the field has fully reached maturity and that we know all there is to know about it. The only thing left is simply added complexity to slightly tweak our understanding here and there. So one can say at that time that we have understood conventional superconductivity.
So in physics, the criteria to say that we understand something is very, very strict. It requires a well-verified theory that matches practically all of the empirical observations, and a general consensus among experts in the field that agree with it. This means that in many instances, physicists would tend to say that we don't understand so-and-so, because there are many areas of physics that haven't been fully answered, verified, or have reached a general consensus. To us, this does not allow us to say that we have understood it. But it certainly does not mean we know NOTHING about it.
This is certainly the scenario for the study of the high-Tc superconductors, where similar phrases have been mentioned. Many in that field would say that after more than 20 years since the discovery of this type of superconductors, we still don't understand them. Now, what does that mean? Do we know NOTHING about them? That is not right. We certainly know A LOT MORE about their properties and behavior since we first encountered them in Bednorz and Muller's paper. And while there are no consensus on the mechanism that causes superconductivity in these materials, we certainly have several possible theoretical candidates. We just haven't had enough experimental observations to be able to clearly say which one is the correct one. But this would be an example where someone could easily say "No one understands high-Tc superconductivity", but it doesn't mean we know nothing about them!
The same can be said about QM. At the most fundamental level, there are various interpretations and attempts at trying to "explain" QM and its mysterious ways. This is what made Feynman said what he did, that the QM equations and description of nature isn't as familiar as our classical world, and that certain things simply have no more fundamental explanation for they to happen. However, beyond that, the formulation of QM is extremely well-understood. After all, even Feynman himself HAD to understand SOMETHING for him to be able to use QM and formulate a whole new branch of QM called Quantum Electrodynamics. You simply can't do something as sophisticated as that without understanding the things you used to get there!
Do we understand QM? Damn right we do! Do we understand it COMPLETELY? Sure if what we mean by "completely" only includes things that we can test and measure. QM is THE most successful theory of the physical world that human has invented up to now and no experimental observation so far has contradicted it. So that alone is a very strong argument that we DO understand QM. However, if we ask if we understand how QM comes up with all the correct predictions of what nature does, or if there's anything underlying all the QM's predictions, then no, we don't. Such a question ASSUMES a priori that there ARE "underlying" mechanisms beyond QM, very much like Einstein's "hidden variables", and that these mechanisms can be observed and measured, something that is strictly required in physics. If there are no such mechanisms, or if they can't be observed, then the whole assumption is MOOT!
I've been known to reply, whenever I get another question such as this, that we understand QM MORE than you understand your own family members. Why? I can use QM to make QUANTITATIVE predictions, not just qualitative ones, and make these predictions uncannily accurate. When was the last time you can do that with your family member consistently, day in, day out, a gazillion times a second? We use QM to do that and more!