If you're cooking low and slow, with enough liquid, you'll end up with a style of stew called ragoût. The trick is to not add the vegetables during the cooking, unless they're either something that you want to break down, or you've added enough acid to the cooking liquid to prevent onions and potatoes from fully disintegrating.
Personally, I prefer to take pot-roast to this stage, as it's what I grew up with.
You might not get the 'cooked to rags' quality of the meat, as you may not have agitated the meat before cooling. In the Good Eats episode 'Stew Romanace', Alton Brown cooks the meat, then lets it cool so it'll firm back up. It's possible that this might happen in your case (although in his, he drains the liquid before cooling).