Jobs and stuff
When I applied I did as many as I could before the deadlines, which I think was a good bit more than six (a dozen?). If I remember correctly they are free to apply anyway.
I somehow managed to download all the applications for one program (they had left the root of their web server unprotected!) and there were hundreds of applicants, all very well qualified. That being said, different programs are looking for different kinds of people. I did Wash U, as did another Amherst student a few years ago. They seemed to be very interested in Amherst kids as opposed to tech school kids, though that certainly won't be true for all the programs. I had a great time there, so I would recommend that one very highly.
In my results I seemed to have better luck in applications to which I could very specifically talk about my interest in working on a particular project(s). Keep in mind these programs double as recruiting ploys for the grad school, so professors are not just looking for summer researchers, but potential grad school recruits. It is a balance though, you don't want to write yourself into a corner where you will only be considered for a single project...
Another thing to consider is that there is, to some extend, a level of prestige associated with certain programs. I think you only need to concern yourself with that in so much as making sure you are not applying to only "prestigious" programs, as there will be more competition for those. The goal is to get research experience under your belt, and any place will do.
In any case, it really isn't about the number you apply to, but how you apply. Make sure you can show how you are personally investing in the particular projects/subject area you are applying to. You shouldn't need to overdo it too much, you are a senior and will be highly sought after for that :)