When I received the TV I did a quick test and it seemed fine, but turning it on this time only displayed the Polaroid logo on boot and then the display seemed to power itself off, even though the blue LED showed that it was still "on". Also, it seemed to pick up some analogue broadcast via the RF cable and produced a good sound output.
A quick Google indicated that this model (and many other Polaroid TVs) has serious problems, particularly relating to bad quality electrolytic capacitors in the power/control boards. While watching a series of videos on Youtube showing how to identify and replace the dodgy caps, I went back in and booted the TV again, this time repeatedly hitting the menu button on the remote. Surprisingly, it went from the logo to a blue screen with a working OSD.
Sounds like it will eventually fail, with all the bad reviews, but for now it seems ok. If you're seeing the same symptoms, see if this works.
Update: Problem not solved - came back again the next day and no amount of button mashing will help it. Also, tilting the screen backwards or forwards causes the power LED to flicker and go dim, which is disturbing. Looks like it's a painful capacitor replacement job which is somewhat likely to fail anyway if it turns out to be another problem.
Poor show, Polaroid!