incompatible observables
Two observables are incompatible when their operators do not commute. There is then no complete common eigenbasis, so a generic state cannot be a sharp eigenstate of both at once, and nature forbids them from simultaneously having definite values in general. Position and momentum are the famous example, and a component of spin along one axis is incompatible with the component along another.
Incompatibility is the deep reason behind the uncertainty principle. It is not that our measuring devices are too clumsy to pin both quantities down; it is that a state with a perfectly sharp value of one observable is necessarily a broad superposition of eigenstates of the other. The spread is built into the state itself, before any measurement is attempted.
Measuring incompatible observables in sequence shows their tension plainly. Measure position sharply, and the state becomes localized; now measure momentum, and you get a wildly unpredictable result, because the localized state contains a wide range of momenta. Measuring the first observable again can give a different answer than before — the act of measuring the second has genuinely reshaped the state.
Non-commuting operators share no full eigenbasis, so a sharp value of one forces a spread in the other.
The uncertainty here is fundamental, not a disturbance story. While measuring one quantity can disturb the other, the deeper point is that a state simply cannot possess sharp values of both at once — the limit holds in principle, even for an ideal measurement.