QM is an intrinsically* probabilistic theory. So if the system is in a generic (non-eigenstate) state, you do have information about any specific observable A, but it is a distribution of possible values.
Repeated experiments to measure that observable on identically prepared initial states will yield a spread of values.
The Heisenberg-Robertson uncertainty principle says that if you choose your initial state to minimise your uncertainty (standard deviation) in observable A, then you will end up increasing your uncertainty in an observable B that does not commute with A.
(*The probabilistic element of QM is not as in classical statistical mechanics where particles have definite positions and momenta described by deterministic Newtonian mechanics. In QM there is no underlying local hidden variable theory --- this possibility has been ruled out by Bell inequality experiments).