Only if the measurements align. If they do you get perfectly correlated numbers. If the angles are 90 degrees apart you get completely unrelated numbers.
The problem comes in when the angle between your two measurements is anything else. The chance that the measurements match is based on the cosine of that angle. There's no way for this to happen if the measurements are independent.
If you try to write two equations, where the first equation takes the secret particle state and first angle and gives you 1 or 0, and the second equation takes the secret particle state and second angle and gives you 1 or 0, you won't be able to reproduce the odds you get in the real world. Only equations that know both angles will work.
