Is it possible that quantum theory is an inadequate explanation of the unobserved, underlying system?

OPINION:  QM does not violate Cause and Effect.  The problem is at that scale the Uncertainty Principle makes it impossible to have complete knowledge.  (We measure the location of a quantum with a high energy particle which screws up our momentum measurement.  We measure the momentum of a quantum with a low energy particle which screws up our position measurement)  Without complete knowledge things become probabilistic.

http://en.wikipedia.org/wiki/Uncertainty_principle

Example:  You just saw your friend walk in the gym with a basketball 5 minuets ago.  Where us he now, what is he doing, why? You have a second friend with a cell phone in the gym, are allowed to ask one question about the current information.  

Where is he?  (locker room? bleachers? basketball court?)  Locker room (You don't know why)
What is he doing? (washing his hands, peeing?  pooping?) Washing his hands (you don't know if he is still there)
Is he still there? (yes, no) No (You don't know where he went)
Where did he go? (basketball court, bleachers, coach's office, I don't know) Coach's Office (You don't know why)

etc

There is a reason (cause) for each thing your friend does (effect).  But, you cannot read his mind and you can only find out one additional piece of information about the current state of your friend.  By the time you ask the second question everything has changed and what you found out before is useless for what is going on now.

Is it possible that quantum theory is an inadequate explanation of the unobserved, underlying system?

No doubt because QM does not address the elements (the innards if you will) of the system.  It addresses only the input to the system and the results produced by the system, the output if you will.  What goes on in that little black box we call the system is mostly a mystery.

That's why we work problems like, for this amount and kind of input, the probability that we get this output is such and such.  We know that we will get a distribution of time and energy outputs, or location and momenta outputs, for example.  And we also know the tighter the spread on one, the looser will be the spread of the other.  That's the Heisenberg.

The causes and effects that swirl around in that little black box we call the system are unknown and probably unknowable, at least at our stage of technology.  But not to worry as long as we continue to get the right answers about the system's output.  And we do, to remarkable precision.  Which is why we don't get too upset that we have no real clue what's going on inside the system.

As to cause and effect, I personally believe there is always a cause for every effect.  But at the quantum level, we are incapable of identifying the causes... now.  Perhaps in the next decade or two, we'll be able to actually follow through and/or model the cause and effect chain of events that result in those outcome distributions we can already predict. [See source.]

Yes, QM is based on statistical, not a causal, sequences. And, as you acknowledge, it is a remarkably accurate theory that has been tested more than any other theory, and still stands, at least 60 years on. This is not, of course, because we cannot measure accurately, or because we disturb a system by measuring it, but because of the imprecision that is part of the wave aspect of reality: waves do not have precise locations.

In physics, we assume that EVERY theory is ultimately an inadequate explanation, and that a better one will be found.

With regards to QM specifically, we do not have a 'nice' conceptual model - nothing that allows us to grasp the nature of this reality with our 'conventional' minds.

It has been suggested that we may not be intelligent enough to grasp the underlying mechanisms of QM.  There is no "law" that says that some aspects of the universe might not be complex far beyond the complexity of General Relativity, which even very few physicists have a deep grasp of. And the fact that the observed facts of QM "boggle the mind", could be a hint that that is exactly the case.