Bob was absent yesterday and has to make up the test. Jane didn’t do so well and you want to let her have a second attempt. You’re only giving students 10 minutes for the quiz today, but Chris just needs a few extra minutes.
What you need is a user override. Not sure when this was added to Moodle, but I found it while setting up my Moodle 2.7 server for next year; and it looks like something that will get a lot of use.
The basics are that you can give a user, or a group, slightly different restrictions on a Moodle quiz.