How common is it?

I'm an NP in an HIV clinic and we don't have any after-hours support. Instead, the answering machine says something like "if this is an emergency, please hang up and dial 911".

The problem this creates is 20 voicemails for you the next day, or a patient who calls and says "I'm sick", hangs up, and then you can't get hold of them. The answering machine provides CYA and little else. Unfortunately our clinic can't afford much more.

Nope. The clinic answering machine says to call 911 or come to the ER.

When I worked at a physician's office, the after-hours call was split between the doctor and the two nurse practitioners. The nurses themselves never took call, although we carried a pager (we attended births with the docs/CNMs) and occasionally the answering service, for some unknown reason, would route the call to us.