I think this is fundamentally a question of interpretation of quantum mechanics. We (society/physicists) do some set of experiments, and come up with a model to explain all of the resulting measurements. Quantum mechanics is a model that explains almost all of the phenomena that we observe. However, it is fundamentally only a description of our observations.
To answer the question directly, in quantum mechanics, a state (of some particle, or collection of particles, as in the blog post) is represented as some vector, and we represent "operators" (which can represent position, momentum, spin, etc...) as matrices acting on the state vectors. When we do a measurement, the state is projected onto an eigenvector of the operator. If we do multiple measurements we have to project multiple times, along different bases. This process is not necessarily commutative. If I measure position and then momentum I will get a fundamentally different result than if I measure momentum and then position.
If we make measurements of two observables that have the same basis, then the two matrices will commute, and there is no such limitation. However, with non-commuting observables this is a fundamental limitation, no matter how good your measurement is, you will always be projecting the initial state in different ways depending on how you measure the different observables.
To answer the question directly, in quantum mechanics, a state (of some particle, or collection of particles, as in the blog post) is represented as some vector, and we represent "operators" (which can represent position, momentum, spin, etc...) as matrices acting on the state vectors. When we do a measurement, the state is projected onto an eigenvector of the operator. If we do multiple measurements we have to project multiple times, along different bases. This process is not necessarily commutative. If I measure position and then momentum I will get a fundamentally different result than if I measure momentum and then position.
If we make measurements of two observables that have the same basis, then the two matrices will commute, and there is no such limitation. However, with non-commuting observables this is a fundamental limitation, no matter how good your measurement is, you will always be projecting the initial state in different ways depending on how you measure the different observables.