threat fatigue
n. Ignoring or downplaying possible threats because one has been subjected to constant warnings about those or similar threats.
Over all, the decision against a public alert represents a significant shift in the thinking of senior government officials over the last several months. Some officials said they had become concerned about the process after several public warnings were issued last fall based on hazy intelligence reports that offered no guidance on how anyone could respond.

They said they feared that such alerts might be causing "threat fatigue" among Americans, who have been bombarded by so many unspecific warnings that they no longer arouse much concern.
—Don Van Natta Jr, “New F.B.I. Alert warns of threat tied to July 4,” The New York Times, June 30, 2002
There's benefit to reminding America of the longtime implications of this challenge. A greater challenge is how we deal with the information we want to share with America, because there is a concern about threat fatigue.
—Tom Ridge, “Interview with Alexis Simendinger,” The National Journal, June 08, 2002
1991 (earliest)
Tales of soaring crime rates cause the old and frail to change their lifestyles to their own detriment, because no one points out that most violent crime occurs between male youths in the vicinity of licensed premises. All dogs of certain breeds are classed as potential mutilators, despite the innocence of most and appalling guilt elsewhere. At some point along the way between global warming, salmonella, Rottweilers, BSE, sewage in the sea, corgis, nitrates in water and supermarket trolleys in canals, danger degrades into farce.

I can't help but think that threat fatigue might partly account for the current 2 per cent showing of the Green Party in the opinion polls.
—Jeremy Burgess, “Danger? What danger?,” New Scientist, October 19, 1991
Of the several dozen media citations I found for "threat fatigue," all but five appeared since June 30, when the F.B.I. warning about a possible July 4 terrorist attack was leaked to the media. Interestingly, the phrase was previewed by Homeland Security Director Tom Ridge in an interview earlier in the month (see the second citstion).