In case of an accident, please call a priest's voicemail

At my parish we are told that we should not expect a priest to be available when a parishioner is terminally ill or dying. Yet The New York Times reported regarding Senator Edward Kennedy that

the priest at the Kennedys’ parish on the Cape, made regular visits to the Kennedy home this summer and held a private family Mass in the living room every Sunday.

and when

...the senator’s condition took a turn ... a priest ... was called to his bedside.

How does the Diocese of Fall River manage to do this for everybody?

(via Terry Mattingly at Get Religion)