There isn't that much of a problem with getting a car like you are proposing, but I would know that all of the following have either been or are very soon, likely to need repacing. Water pump, radiator, alternator, head gasket, battery. The problem is that usually when you are buying on the cheap it is because you don't have a lot of bucks, so be aware. I just bought a 94 318i with 80,000 Kilometres (50,000 miles) on it and within a week two items were wrong with it. The alternator went the day after I drove it off the BMW lot (the dealer replaced it for free) and I discovered a slow leak when replacing the muffler - mechanic caught it when the car was raised - from the radiator. Since I have been frequenting these forums for a while, I knew radiators were an issue on the 318i so I had it immediately replaced rather than wait for the break down I knew would eventually happen.
No matter what car, predictable items are going to need replacing; I know as I've replaced just about everything in cars multiple times, except the engine. I was almost there on the last car, but got rid of it.
If you have only $4,000 dollars to your name, your better to really look for an older but useable Bimmer for $2,000 (they're out there) then right away replace items to burn up the remaining money. Assuming you don't have the records, I replace the radiator (this almost gives you a free coolant purge), the alternator, the water pump, and the belts if they were old, and hoses if they look bad.