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Yes, that's exactly why your notifications don't pop up, because you're killing the app that's the source of them.
A Task Killer isn't a necessity for Android, but more of a "nice-to-have-if-things-start-going-to-hell" kind of deal. Don't rely on it. Don't run it 24/7 (my most-killed task is the task killer itself, honestly). And most importantly, be smart about what you kill.