The blower final stage fan resistor is known to fail. Symptoms are the heater fan speed varies regardless of the setting, the fan will not turn off when the car is switched off, or the fan will not switch on at all.
Car Problem Reports
BMW 330i Blower Final Stage Fan Resistor May Fail
BMW 330i Problem
Average Mileage: 118,571 mi (95,000 mi - 134,000 mi)
My heater/Air conditioner blower has always blown intermittently, and at varying speeds. 3 days ago the blower stoped working altogether. I can still feel heat from the vents, but is not blowing out and causes quite a problem in the morning when the windows are foggy, will not defrost.
Visitor, 2001 BMW 330i, 134,000 mi
Blower Final Stage failed around 70k
Visitor, 2002 BMW 330i, 95,000 mi
Stop and go on the blower for a few days before failing completely.
Visitor, 2001 BMW 330i, 126,600 mi
the blower will not turn on at all