How much can one download from databases before it is too much? According to this article, a student was arrested for downloading ~4 million articles from an on-line database.

I want to review all existing articles in my interest area before writing my paper, so I need to download and skim through several thousand articles. I read the terms of service of my school's database, but it just says I should avoid trying to make a "collection".

At what point does the downloading become enough that the databases start to complain? 1,000? 10,000? 100,000? 1,000,000? Do they send out warnings?
According to this article, a student was arrested for downloading ~4
  million articles from an on-line database.


That was an extreme case. He had history of downloading and releasing large amount of information. In this JSTOR case, I don't think he had actually released the articles, although the indication was strong. He was charged with "unlawfully breaking in a protected computer," which I guess he probably downloaded the paper with unusual means, perhaps using codes or hacks. Overall, the whole case was very politically charged and I will not use that as a benchmark in your case.


  I want to review all existing articles in my interest area before
  writing my paper, so I need to download and skim through several
  thousand articles. I read the terms of service of my school's
  database, but it just says I should avoid trying to make a
  "collection".


When there are several thousands of articles that are pertinent to your research interest in ONE paper, I would suggest you to refine your research interest. And no, you don't need to review all existing articles, just pick the important seminal works, and then trace the major works along a very well define research interest. You can also use a few prominent modern works as seeds and use their reference list to snow ball your library.


  At what point does the downloading become enough that the databases
  start to complain? 1,000? 10,000? 100,000? 1,000,000? Do they send out
  warnings?


You should be able to download as much as you want as long as you follow the instructions and rules. For large download volume, I'd check with the librarian to clarify:


What the institutional limit is. There may be a download limit and if you'd get a warning letter it's likely from your school because your thousands of download could have cost them a lot more than usual. Some school libraries may ban accounts associated with a large download volume as well; it's better to give the librarian a heads up.
Your student account payment scheme. Some schools may have limit and beyond which the students may have to pay. I know that this is certainly possible for inter-library loan (which you may have to do if a few thousands paper is your goal.) A peer of mine got billed for more than US$500 because she ordered about a hundred papers through inter-library loan.