Unfortunately, any car can break, and cheap cars are more likely to. I had a cheap Civic that cost an arm and a leg to keep running, while also having an even cheaper Chevy Citation that went over 300k miles with almost no cost at all (just gas and the occasional, and I do mean occasional, oil change). you can scope out a car and carfax it, and have a mechanic check it over, but still have a catastrophic breakdown the following week.
And of course, geographic location will have a lot to do with it. In the midwest and northeast, cheap cars will tend to be weathered and very likely rusty, needing repairs to the undercarriage way before the engines and trans wear out, while on the west coast and in the southwest, they are more likely to wear out mechanically long before they rust away.
Where are you located? That will make a big difference on what is available in that price range.