The word "simultaneously" is there because you can obtain a precise measurement of an particle's position and afterward obtain a precise measurement of the momentum. But you can't do both at the same time. Further, when you make the second measurement (the measurement of the momentum), you disturb the object's position, so it in fact no longer has a definite position, and the position measurement you had made at first no longer contains any meaningful information about the particles position.
For a concrete example, consider a particle in the ground state of the quantum harmonic oscillator. In this state, $\Delta x$ and $\Delta p$ are both non-zero and constant in time, so neither the position nor momentum is known. Now if you measure the position, the particle's wave function collapses to a delta function, and you observe a precise position for the particle. But now when you measure the momentum, the wavefunction collapses into a plane wave state, which has infinite extent. So while you obtain a precise value for the momentum, you have lost all the information about the particle's position.