I bought an 87 325i convertible with 140K miles in january of this year. I *love* it and can't get enough of driving it. BUT, there were significant problems and I have since turned into an amateur mechanic in order to deal with them. It's been great fun for me, but make sure you know what you're getting into. I think my car was driven really hard in the past...but here's what I've done to it so far:
replaced intake manifold gasket (vacuum leak into the engine)
replaced front shocks, they were totally gone.
resealed oil pan -- I was eating a quart of oil every 275 miles. that fixed it pretty good.
replaced inside driver's door handle, it cracked.
tune up...oil...filters....spark plugs...carb cleaner...
replaced front brakes
replaced many little plastic pieces that you can only get from BMW for very high prices
replace axle CV boots, they're ripped (very bad but aparently its rare for them to rip, but check em out before you buy)
replace both front wheel bearings (loud grinding noise while turning)
i bought the car for $2600 originally, and have dropped another $4500 on repairs/tools/parts. All in all I'm still really happy for it. Hope that helps.