Why do planes above FL290 need more vertical spacing? At altitudes no higher than 29,000 feet, planes need only 1,000 feet of vertical separation. 
However, planes over FL290 need twice that, i.e. they need 2,000 feet of vertical spacing between them and a plane with otherwise insufficient lateral separation. 
Why is the extra vertical spacing needed, and why is it needed over FL290 specifically?
The atmospheric pressure changes more rapidly with altitude near the ground than it does higher up.  Assuming an instrument has some error based on determining the difference between two pressures, this error increases with altitude.

Using a standard atmosphere, the pressure difference between 3000 and 4000 feet is 3.3 kPa.  The difference between FL290 and FL300 is 1.4 kPa, less than half.