I went down the bangernomics (ish, it wasn't quite under a grand, but it was a nice big family car in v good nick for its age) route a couple of years ago, and it died just before Christmas of a snapped cam-belt/water pump/valves.
Luckily it happened quite close to home, but it got me thinking that it could have happened at any point in the previous 2 years, with wife and kids in the car, and potentially a long way from home.
Decide if this is what you'd be cool with. Potentially stuck (or at least waiting for Mr RAC to turn up) a long way from home, dog/kids in there too, and faced with no transport when you get home. Sounds negative, but it's the risk you take.
So I had a good look around and found a 7 yet old Mitsubishi Lancer 1.9 diesel (so very reliable if boring), 70,000 miles, 1 owner from new, FSH, and still on extended warranty. Paid £2,600 (proper bargain IMO) and I'm a lot more confident that I won't get stuck somewhere. YMMV!