I personally try to buy cars with less than 45,000 miles on them. That gets you 1) hopefully newish (not factory) tires and 2) a few years before anything major could go wrong.
For a few thousand dollar budget, I don't think you are going to find anything that is worth buying. I would hate to "trade" a car payment for a less reliable car and spend more monthly on repairs. It's the known vs the unknown.
I bought a car with a salvage title several years ago. It had been fixed and worked fairly well (just a few minor problems that DH fixed). It's one tough car and has served me well. Most people wouldn't recommend this but If you know a good mechanic that has access to totaled car... You can get some decent deals.