Gareth,
Thanks so much for your help. You correctly diagnosed the problem as related to a duplicate record in the database and directed me into the correct area in your responses in Tracker http://tracker.moodle.org/browse/MDL-31441 I did the search you suggested, but I still did not see duplicates. So, I took a more radical approach, based on what you said.
I edited the grade_object.php file that was generating the error message and changed line 161: "if (count($instances) > 1 {" to a 2. Then I went back to my site, tried to enter the quiz again, and it opened! Now I was able to deleted all attempts. Then back to line 161 and changed it back to a 1. Now, all works. But this still doesn't explain why it happened.
I am going back into the database to look for whatever I might find. Also will review my questions, their scoring, and so on. Also made a backup of the quiz as a .mbz and may delete the current version and re-import it.
Again, thanks for all your guidance on this. I will also follow the bug fix to see where it goes. Seems like this is something that should not be able to occur.
Rod Wolford