This is to help anyone else who has the problem we had with a generic Android tablet, and games that store the progress ‘in the cloud’ – like The Simpsons Tapped Out.
After installing this game on a new, generic Android tablet running Ice Cream Sandwich, the game started at level 15 with someone else’s data.
It turns out that the ‘unique’ identifier “Secure.Android_ID” wasn’t as unique as it should have been, apparently not an unusual problem with small manufacturers’ ROMs.
Note that changing the device ID could screw up your installed apps, lose your existing game data, make you ill or set fire to your pants (in approximate descending order of likelihood). Do this at your own risk. You know you will anyway.
It’s possible to change the Android_ID value (note – not the telephony ID, which is related to phones’ IMEI numbers) providing the tablet is rooted.
I installed and ran Root Checker Basic and found that the tablet was supplied rooted.
Then to make sure the new id was picked up by the game, I uninstalled it, rebooted the tablet, deleted the data folder for the game from Android/Data using a File Explorer like this one, and then reinstalled the game.
I hope this helps someone else in the same position. Or me again at some point in the future.