Hi Jacob -
We're going to need a lot more information to help. First, what version of questionnaire are you using? You can find this on the plugins overview page in your administration menu. Second, what version of Moodle are you using?
What types of questions does your questionnaire use?
When you say the system is not saving their answers, have they clicked the "Submit questionnaire" button? Or do you have the questionnaire set to "Save and resume" and they are using the "Save" button?
mike