In fact I received a call from an unknown number (that Google later showed as belonging to a local hospital) and the caller did not leave any message. A few days later someone called again and did leave voicemail telling me that a very good, old, friend was in the hospital and had left my phone number as a contact. He died several hours later.
While responding to the initial call would almost certainly have made no difference to my friend's survival, I wish that I had been alerted sooner. Just a "this is Dr. X calling about Mr. Y" would have been fine (and in fact was the content of the voicemail left in the second call).