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.
The separation rules you mentioned (1000 ft < FL290, 2000ft > FL290) were inplace due to the reduced accuracy of old altimeters at these altitudes. A less accurate measure meant larger safety margin was required.
Today, most if not all modern airliners are certified for RVSM (Reduced Vertical Separation Minima). These aircraft are equipped with more accurate altimeters and ADCs (Air Data Computers). RVSM reduces the required separation from 2000 ft to 1000ft at FL290 - FL410. In case of malfunctioning RVSM-related equipment (included in the MMEL) that aircraft must avoid RVSM airspace (= fly lower than FL290) until this equipment is fixed.