Absolutely more than available for £4000 now even through places such as autotrader.
This sort of money wont get you a facelift Sport as Fox, Peter, and Olly (amongst others) have and post on here, but it will get you a nice sensible mileage example of a prefacelift SE with plenty of options. You can update the asthetics later too with new lights and even the sport bodykit, as the body is the same.
Get a 6 cylinder pre facelift SE, with leather and as many options as you can find. Cars to go for are the 523i and the 528i depending on how your insurance fares.
The older 520i is only a 4cylinder engine and although reliable, its not hugely powerful and is eclipsed by the much better 6cylinder models. Running costs between 523 and 528 will be largely the same.
Remember that any cars you will be looking at will be getting on for 9-10 years old, and thus they WILL break and they WILL cost you reasonable money to keep running. Prices that the other guys seem to quote for repairs seem quite reasonable however, there are loads of specialist BMW places which will carry out repairs relatively cheaply. Just dont expect Ford Focus running costs, these are a much heavier more substantial car, suspension components will need doing, tyre wear will be heavier, fuel consumption will be higher.