Who may be anointed? This is the question pastors and pastoral care ministers have attempted to address since Vatican II. The sacrament formerly known as “extreme unction” was delayed until the time of death. So there were a lot of cultural expectations to overcome.
8 The Letter of James states that the sick are to be anointed ill order to raise them up and save them. (See Council of Trent, sess. 14, De Extrema Unctione, cap. 2: Denz.- Schön. 1698) Great care and concern should be taken to see that those of the faithful whose health is seriously* impaired by sickness or old age receive this sacrament. (Sacrosanctum Concilium 73)
The starred word “seriously/periculose” has the following note:
<<*The word periculose has been carefully studied and rendered as “seriously,” rather than as “gravely,” “dangerously,” or “perilously,” Such a rendering will serve to avoid restrictions upon the celebration of the sacrament. On the one hand, the sacrament may and should be given to anyone whose health is seriously impaired; on the other hand, it may not be given indiscriminately or to any person whose health is not seriously impaired. >>
The Church counsels clergy to exercise judgment, and ask a doctor “if necessary.”
A prudent or reasonably sure judgment, without scruple, is sufficient for deciding on the seriousness of an illness; (See Pius XI, Epist. Explorata res, 2 February 1923: AAS 15 (1923) 103-107) if necessary a doctor may be consulted.
Other considerations are outlined in sections nine through twelve.
9 The sacrament may be repeated if the sick person recovers after being anointed and then again falls ill or if during the same illness the person’s condition becomes more serious.
10 A sick person may be anointed before surgery whenever a serious illness is the reason for the surgery.
11 Elderly people may be anointed if they have become notably weakened even though no serious illness is present.
12 Sick children may be anointed if they have sufficient use of reason to be strengthened by this sacrament.
In your parish, do you find these reasons are fairly well understood? Do people ask for anointing before cancer surgery, for example? Are children anointed or is this an adult sacrament in your parish?
Catholic Sensibility 
The illness need not be physical – it can be psychological as well. This is often lost on both priest and people, who operate on preconciliar autopilot in that regard.
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Not just priests, “who operate on …autopilot,” even newer priests, maybe less informed, such as those who come from different cultural backgrounds and have difficulty understanding or expressing things in the English language have a tough time.