Several times now I have tried going through saarthal, but the warning from the psijic order almost never comes. Either nothing happens, Tolfdir freezes and nothing happens, or in one case the psijic order member spoke to me and then talking to Tolfdir didn't advance anything.

Clear your XBOX 360 cache, load a previous save and try again.

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At the Eye of Magnus quest, if a Psijic Order is already present in the hall, skipping over the cutscene between Ancano and Tolfdir, prevents Tolfdir from finishing his cutscene.

This makes the quest impossible to finish. You must reload a previous save before entering the college to continue.