So angular momentum is conserved about a point if no external net torques act about that point. But is there any occasion when this is only true about certain points? In other words: can it happen that angular momentum is conserved about some points in a system but not others?
As you already said, angular momentum about a point is conserved if and only if the net external torque about that same point is zero. This is always true.
It might happen though that the net external torque is zero about one point and non zero about some other. In this case angular momentum is conserved about the first point whereas it is not about the second. As an example, consider a particle in free fall. The torque due to gravity about any point in the vertical line containing the particle vanishes and therefore the angular momentum about that point is conserved. For the same system, consider a point belonging to another vertical line which does not contain the particle. The torque is non zero so the angular momentum is not conserved.