Hi, I have a 6 switch Atari 2600, NTSC model that's mostly been sitting on the shelf. I'd use it periodically though had some trouble as the power jack was loose, and it wouldn't power on unless the AC adapter was plugged in at a specific angle (have had a similar problem with my genesis model 2). Eventually however, I got to a point where it wouldn't power on at all. I did clean the contacts to the RF cable to eliminate the possibility that it was a video connection problem.
I suspect the issue is that the jack is loose, and either needs to be replaced or resoddered. I had been using a 12 volt AC adapter not realizing the system was meant to have a 9 volt adapter. Not sure if a 3 volt difference would have caused a problem.
Google hasn't been particularly helpful and I'm almost always redirected to the forums whenever I search an Atari problem. Not sure if this is a common problem, or if there's any recommendations beyond just replacing the power jack.