What factors influence the maximum tire pressure other than the tire itself?
The usual cause of inner tubes exploding while being inflated is that part of the tube is pinched under the rim, or was damaged by a tyre lever while you were fitting the tyre (but manufacturing flaws can happen - I once had a puncture in a tube next the stem where it was difficult to patch, and my spare tube failed at the same place as soon as I inflated it (and there wasn't a sharp edge on the hole in the rim)).

But apart from that, if the tube and tyre are properly seated, and the tube is the right size, the tube shouldn't limit the pressure.

Rim tape can make a difference if the wrong sort is used, but shouldn't otherwise, see http://www.schwalbe.co.uk/_webedit/uploaded-files/All%20Files/Technical%20Info.pdf and search for "Which rim tape should I use".

Tyre sizes and rim sizes vary slightly, and a combination that is a slightly loose fit will let the tyre blow off at a lower pressure. The label allows for that, at least to a certain extent (see http://sheldonbrown.com/tires.html#pressure and scroll down to "Pressure Recommendations").

Using a tyre on a rim much narrower than it is intended for can make the rim fail and let a tyre blow off: http://www.sheldonbrown.com/tire-sizing.html#width. So can using a badly worn rim, of course.