Basically, hidden variables models can't explain the results of some of our experiments and no-one's found a way to make the two match up.
Reference: https://en.wikipedia.org/wiki/Bell's_theorem
It's more accurately phrased as, "No physical theory of local hidden variables with observations that are statistically independent can ever reproduce all of the predictions of quantum mechanics."
This is actually a new loophole that 't Hooft is exploring in his cellular automata interpretation of quantum mechanics, ie. it has local hidden variables, but because all matter shares a common history from the Big Bang, no observations we can possibly make can be statistically independent of each other.
Yep. When I'm reading these articles, "Everything Connects" is a pretty useful heuristic for flowing with apparent discrepancies in theories...although of course nailing down concrete theories and finding precise explanations for phenomena is important.
"Thus, it is conceivable that freedom of choice has been restricted since the beginning of the universe in the Big Bang, with every future measurement predetermined by correlations established at the Big Bang"
> "No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics."
GIT paraphrased to match: "No axiomatic system can produce all truths within that axiomatic system."