Increasing time on slow cooker recipes
It depends on the particulars of the recipe and the ingredients and the slow cooker itself.  

You say you are using chicken or beef, but are the pieces of meat small or large?  Is the meat lean or fatty?  I've had problems with overcooking large pieces of lean meat, such as chicken breast, but smaller pieces and/or fattier pieces seem to hold up better to extra-long cooking.

Another issue is proportion of liquid.  For example, soups tend to work better in your scenario than "drier" recipes.

And, of course, the slow cooker itself can make a difference.  I assume you are talking about 9-10 hours on low, not on high.  But low itself is not likely to be entirely consistent between cookers.  For example, a large slow cooker is likely to put out more total energy than a small slow cooker, so for extra-long cooking times, you are probably better off using a small cooker with a full pot that a large cooker with a half-full pot.