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PhilGoetz comments on You can't believe in Bayes - Less Wrong
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Sure. As I noted in the original post and also in response to your comments, sometimes it's fine to do this. Sometimes it isn't.
You are throwing away information when you threshold, period, end of that discussion. If that information was important, you made a mistake. It is easy to identify these situations because they have aberrant expected values in low-probability outcomes. And it is easy to avoid them, unlike with tigers.
If the 'information' was noise, then it's not information, it's just data. Whether it counts as information depends upon relevance, which is partially what the threshold is for.
Link for more info
Of course, this is a jargon use in information theory. But that seems like the relevant domain.