I think one needs to have the right expectations about what they want to achieve from an accelerator. If you are a company with a broad audience like consumer web or consumer mobile, an accelerator will help you build your company fundamentals, guide you through the generic process of starting up and at the end of it all give you audience with the investors; and if you are in a program like YC garner you a little more attention. So I really fail to see how useful it is to go through two different accelerators other than the just the last two reasons.

If I were running an accelerator and you were applying to mine after having gone through another program, I wouldn't just look at if I will gain (financially or otherwise) from you but really question how you will gain from having you in in the accelerator, having already gone through a similar, perhaps even conflicting process.

For a company with a more specialized market like education or health, while the requirements above stay the same, there are important differences. So going through a generalized accelerator to build those fundamentals and then going through an accelerator specific to your industry makes sense because while building strong fundamentals in business may be the smae, scaling up, selling and operating successfully in these industries is rather different and perhaps even more different than the consumer space, like Imagine K12 for education and Rockhealth for health; or even the other way round.