Video files themselves can contain transparent backgrounds, but in formats that are not web friendly, which is the reference for your question. To my knowledge there is not a web standard that natively (and universally) supports transparency - but read on...
The WebM format (VP8 codec) has had some interesting work done in this regard:
Alpha transparency in Chrome - video
Limited, but a start.
What is more interesting is moving away from a format~codec based solution and into an HTML5 canvas approach. This different solution has the benefit that all current video web standards can "keep on keeping on" without a fracturing in standards.
Here is a quick video tutorial that shows a manual (but still pretty quick method) of getting a sequence of frames added to a Tumult Hype document: http://j.mp/1l3yFpr
Yes, but too tedious for anything but a small collection of frames... looking forward to Hype having an "image sequence" import capability. This feature would would expand Hype's design capabilities while being a "good bang for the development buck" (I would think).