Yes. Some game saves (Black, Timesplitters 2, and most EA games, to name a few) are embedded with xbox specific eeprom data.
There was a PC utility floating around the net for resigning these problematic saves to a new xbox but I have never used it, so I can't remember the name or attest to it's success rate*.
Another option is to use configuration magic on the new xbox and manually edit the eeprom data to match the values from the old xbox*.
Another option is to simply dump the eeprom from your old xbox, prepare it with liveinfo, (sometimes required if the 2 consoles in question are different versions), and then write it to the new xbox*^.
It is basically a matter of personal preference which method you use. I have used both of the eeprom methods so I know they work, but I also always hardmod via TSOP flash or homebrew chip, and I don't keep 1.6's around long (softmod and sell cheap
* This method can get complicated if either xbox uses a virtual eeprom (softmod).
^ This method may fail if either xbox is a version 1.6.
NOTE:As you may have noticed from my comments, this task isn't worth the time and effort if either of your xbox consoles (especially the new one) were/are softmodded.This post has been edited by xboxmods2977: Apr 21 2012, 06:18 PM