For example, I'm working on an A4 story, the black background bleeds off all four sides. What I have done is increase the size of the art to 216mmx303mm (3mm bleed on each side) but the document itself remains at A4 size.
Right… this is the bane of the letterer's life because many, many artists don't seem able to grasp this, either. To all intents and purposes, the actual page size (in this case A4) is completely irrelevant if the publication has bleed.
You only need two measurements, and these should come from the printer: the full bleed size, and the safe type (AKA live) area. Your document should be the whole, full bleed size. If the printer says there's 3mm bleed then your document should be 216x303mm and the artwork should sit exactly on it. If it's 5mm bleed then your document should be 220x307mm.
All panel borders, speech balloons,
anything that doesn't want to get cut off by the finished page edge (when your oversize page gets cut down to A4) should go inside the Live/Safe Type area.
You will frequently find that when you fit the full bleed dimensions to the edge of your AI document, the artist has drawn the panel borders outside the Live area. This is not your problem, and there is no easy fix for it. All you can do is ensure that the lettering, at least, stays inside the Live area -- the art will have to take its chances.
Cheers!
Jim