I just realized that our iMac was having a small problem with the the new version of Mac OS X. 10.4.9 now includes improved checking for damaged disk images. All of the disk images that I had created so our kids could play their games, without using the CD, now are being reported as being corrupt. (I doubt that they actually are all corrupt, unless the previous version of Disk Utility created corrupt images.)
Unfortunately, recreating the disk images from the original CDs does not solve the problem. The error message still appears.
The only way that I could work around this problem is by modifying the AppleScripts to not fsck the disk images when they are mounted. For example, the scripts now execute commands like:
hdiutil attach '/Users/Shared/Didi & Ditto.dmg' -mount required -readonly -noverify -ignorebadchecksums -noautofsck