Is it possible to travel between India and China by rail?

India and China both have extensive national rail networks. Is it possible to travel by train/rail between India and China? I have found several articles (e.g. at The Hindu and at India Today) discussing the possibility of an India-China link, which leads me to believe that there is no direct link. Can an India-China rail trip be done, either directly between the countries or by transiting other countries (e.g. Nepal, Pakistan, Bhutan, or Myanmar)? Changing trains is ok.

This question has nothing to do with what visas (if any) would be needed for such a trip. My question is whether such a rail link exists and, if so, whether a traveler could actually book travel along each leg.

Note based on answers so far:

While I specifically said that indirect travel using a third (or even fourth...) country would be a good answer, I think that it should be restricted to using at most two other countries (for example, India->Bangladesh->Myanmar->China), unless a case can be made as to why three or more transit countries would be considered reasonable for a journey. If the only way to go by rail would be a journey involving ten different train changes to Europe and ten back to Asia over a six week period just to get over the Himalayas, we can say that such a journey would not be reasonable and I would accept an answer of "No, this is not possible, because six weeks and 20 train changes" in that case.