Unless the net is directly touching one and some sort of force is involved... some trellis nets are better than others as far as loose threads that can stick to buds etc.
common sense should guide you if you come across something that invovles this question. Pull the net down below a bud if it directly comes into contact or is pressed between net and stem. simple as that. avoid direct touching of flower as best you can to avoid any loose threads. Usually the net underneath bud sites unless it's purely for stability and not a training tool. The nicer bungee cord type of net generally loses fewer threads, if any. if you can see threads loosely hanging from your trellis and don't want to smoke some of those threads, get a btter one.
looks like you got the parachute/bunge cords stuff. so that shouldn't be an issue.
Really thin trellis net could cut into stuff with enough pressure -- just as the cheap garden wire can, but you have the thicker kind.