Automatic Transmission problem

I have a 1990 520i with Automatic Transmission. The car is giving me a bit of trouble, The problem occurs in two different ways, 1. if the car is accelerated quickly the car will start to chug and splutter at about 30 MPH. 2. if the car is accelerated slowly it will reach 80 MPH before the problem occurs. The problem appears to come in at around 5000 RPM. Also the gear changes are not as smooth as you would expect from an automatic.

I have checked the ATF and all looks ok, Any other ideas on how to fix this problem would be greatly appreciated.