I own a 2001 Chevrolet Cavalier, automatic transmission. It has 155,000 miles. I received the car from a mechanic shop. It had been used as a loaner car. The mechanic told me that it has a new block. I have only owned the Cavalier for a few months. Shortly after getting the car home, it overheated on my way to the station. After letting it cool, I drove back home. It became very hot on the way home, so I let it cool down again for 15 minutes, or so. After all this I installed a new thermostat and topped off the coolant. Ever since then, a couple months now, the car has had no overheating. Now, all of a sudden, the Cavalier nearly overheated again on a side street. The thermostat needle fluctuates above the half way point and hot when I drive in city. It no longer settles in the middle of the gauge, as it has been before. Now I took off the radiators overflow cap, when it was very warm, and relieved a good amount of air pressure. I promptly put in more 50/50 coolant mixture. The vehicle was allowed to cool down. Now when I drive in city, the water temperature gauge is running on the hotter side constantly. The new thermostat is a 180° f unit. It only runs cooler on the highway now.
Sounds like the mechanic unloaded his problem on you. Always test drive cars and have them inspected by a third party.
There's a few possibilities.
water pump failing - check by removing and inspecting
plugged up radiator - Check by flowing water through it
head gasket failure - Grab a test kit. Scotty has videos so search his Youtube.
I really would not expect much from a Cavalier though. They weren't great cars when new, let alone 20 years later.
Agree with everything you said. Though, anecdotally, my mother somehow got 300,000 miles out of her cavalier with scarcely more than routine service. There apparently were a few peaches in amongst the lemons. 
@Dad2LM2 With a bit of knowledge, and in the right, meticulous hands, they can last because they were relatively simple, parts were cheap and plentiful, and they're fairly easy to work on. Fit & finish is a bit shoddy, but the powertrains are generally reliable (at least until recent years). Unfortunately they tend to get bagged on by youngsters.
I appreciate all the help with my 2001 Chevrolet Cavalier. I am still driving it, keeping an eye on the fluid levels. There is a slow coolant leak in the system I have yet to locate. So occasionally this winter I have topped off the coolant reservoir. I installed a front wheel bearing a couple weeks ago. Otherwise vehicle is ok. I was on the highway recently driving along at 65 m.p.h. and the rpm seemed to stall downwards for just a split second- a few times on my way home. I put a bottle of Heet into the gas tank, and put a slightly higher octane gas into the tank. It seems to have resolved the hiccups for now...knock on plastic (dashboard).
You might want to consider a new, $12 radiator cap. They probably just put the 21 year old one back on.
