Comment by supertrope

You can thank HIPAA for that. Under the Privacy Rule medical information has to be guarded. While I have seen some practices let you indicate on the patient forms that you allow brief or full voicemail, many won't do it as there's no one to confirm their name and DOB. Even the fact that you are a patient at a clinic can be protected health information (for example getting a call from a women's health clinic or drug rehab center that doesn't block caller ID can be compromising).