It's got nothing to do with offence. Making credible threats is illegal, and for good reason. Making credible threats can have extremely negative consequences for the person or institution being threatened. Do they do nothing and take an unknown risk, which leaves them open to being blamed (and probably sued) if something does happen? Do they cancel their event, stay at home with a loaded gun and put their life on hold? Not to mention the stress which can affect their health.
Defining what is credible is the tricky part. I'm not sure this is, but it's probably enough to warrant an investigation to see if the person had the means to carry out an attack. It's a difficult line to draw.
On the other hand things that are clearly jokes, like the guy in the UK who was eventually cleared of posting a tweet joking about blowing up an airport if his flight was cancelled, should not be investigated or prosecuted. I'm sorry I can't give you the exact limit, but that's the nature of life. Speech is imprecise.