We had a similar problem a little over a year ago. After trying a 30' snake we still couldn't get good use out of our kitchen sink before it would start acting slow again. We ran the snake through two cleanouts. Our drain lines run in a shape of a capital "T". The kitchen at the top left of the "T" and the washer/garage at the top right of the "T". After being quoted $1600 to get it hydro-jeted and video inspection I had thought I'd have to sleep on that amount and rethink the hole mystery of why the snakes didn't work, etc. I decided that I would hook up a new garden hose we'd recently bought for the wife to use (cause the other one we had was one of those hard stiff hose that you have to wrestle with unless the sun has been beaming on it all day). Anyway, hooked up the wifes clean and shiny new hose to the water heaters drain spout and ran the other end down the clean out port on the outside wall from the kitchen sink. This hose was cheap, which made it bend without resistance, compared to the other stiff hose we have. Pushing it in was easy and the usual black nasty water come gushing out at the same rate as the water going in through the hose. I thought to myself, either I'm gonna fix this or I'm written a check for $1600 real soon. So, I said to myself, F it...I'm gonna just let this hose run for half hour if I have to. sliding the hose in and out a few feet at a time as the water coming out became less black, I'd push some more hose into the cleanout. I kept doing this until the water that was coming out was clear...for nearly 25 minutes or so. Then I heard this swoosh sound and the water quit coming out. The drain was flowing like brand new. I'm thinkin to myself, I either opened up a cracked pipe into gopherville, or I'd just saved myself writing a fat check. It's been running great ever since.
I still to this day remember, before the above happen, when I tried caping off the kitchen drain and forcing water through the garage cleanout. We'd capped it off so tight that the water was either gonna pushout through the clog or my vent pipe would look like old faithful. Funny, as hell...it looked like old faithful. That's when I called for quotes. Glad I didn't give up and just kept at it. It makes sense though...keep pushing water in the cleanout until the water comes out clear...then feed in some more. Repeat.
Good luck and let us know what did it for ya.