I don't think price is a good way to judge. I bought my CD player, the Meridian 508.20, for a price far lower than I would have expected to pay for it. Plus, it was (and still is) in absolutely perfect condition without a single scratch. It's much easier establish "price points" for new equipment because the prices don't vary as much, and neither does the risk involved in purchasing. I'd say, though, at around $600-700 you can start getting used CD players that are really leaving the "cheap" category.
Keep in mind, also, how fast digital technology and product offerings change. When I bought my Marantz CD6000OSE, the Sony NS500V was not available, and neither was the NAD C-541i. If I had been more knowledgeable, I could have bought the Sony SCD-333ES new at about the same price ($400) from Oade Brothers -- new. It's like with a lot of products where you think that it should be easy to determine quality by price. Well, you just can't. If you separate by $1000 or even $500, sure, most of the time you'll be getting something better if you choose the more-expensive player. I prefer my Meridian 508.20 to the Cary CD-308 which costs a few hundred more than I paid for my player -- but again, I bought mine used. In a few months the prices will all change, the products available will change, and each prospective buyer will likely have to discard a lot of the information provided by reviewers this fall.
That's why the word "cheap" means so little. If you want to find a group of people happy to give you pseudo-objective definitions of the terms "cheap," "best," "worst," and "supreme," go visit the Audio Asylum's Digital Drive forum.