Yes, but this is now termed a revalidation. Under the European Class 3 requirements you can revalidate your medical certificate up to 45 days before the expiry date of your last certificate. Your certificate will then be valid for 2 years if you are under 40 at the date of examination or 1 year if you are over 40. If you continue to go to your AME for an examination in the month before your expiry date you will see no change. The European Class 3 medical certificate revalidation rules are similar to those used for ATCO licence competence assessments.