Yes it is possible to have Participants change the Event Registration Questions.  Follow the steps below to enable this:
  1. Go to Fundraising > Select TeamRaiser
  2. Find your desired event and click Edit
  3. Go to Step 3. Select Event Options
  4. Click Related Action for Edit Advanced Options
  5. Go to Step 3b. Define Event Options
  6. Mark the checkbox for Option 21. Registration Question Response Editing
With this enabled, there will be a link along the right hand side of the Participant Center homepage titled Update Question Responses where participants can change their answers.