Uncertainty Principle refers to the arbitrary precision of two operators. You can measure each operator individually to arbitrary precision. In theory, that is.
If the two operators commute you can get theoretical arbitrary precision on both measurements together. Since their commutator is zero.
If they don't commute, the product of standard deviations of each measurement operator is restricted to no less than half the expectation value of their commutator. (photon-torpedo's comment at the top of this thread gives a better summary).
The UP is not only: "You can't know position and momentum together." It rather: "Position and measurement _cannot be defined_ together."
In classical mechanics, you can imagine a particle being at a certain position with a certain momentum. In quantum mechanics, you cannot.
That the product of the standard deviations of position and momentum are at least half of h-bar.
It says nothing else about what that "means", the mechanism by which it comes about, or whether it is a fundamentally "true" description of reality.
It's not false, but it's not entirely true. The uncertainty principle doesn't state that you can't _measure_ p and q with arbitrary precision. It states that you cannot _imagine_ them with arbitrary precision. It's like the double slit experiment: when you measure which slit is traversed, interference patterns disappear; but, when you don't measure, you cannot even _think_ that the photon is passing through one slit instead of the other.