Well, it looks like I'm answering myself for a second time, but this might be useful information to others. I'd read somewhere of a similar problem with the X10, that was linked to a corrupted SD card (or from the camera's point of view). It seemed to me that the camera couldn't in some way shut down properly, and maybe that was because it couldn't finalise the writing of the last file, causing the firmware to "hang". So, I tried a new SD card - seemed to fix the problem - I then low-level formatted the SD card in the camera at the time of the problem using Windows (formatting in-camera did not improve matters). As I write this I've taken many shots and turned the X20 on/off numerous times with no re-occurrence. So...maybe that (SD card) hypothesis is correct? Time will tell. Anyway, it's not on its way back to Fujifilm.