Plumber's snake buckles when trying to unclog sink
Well, you found where the clog isn't, at least. However, if you can't go to a lower part of the plumbing or further from the street through another sink, toilet or basement clean-out. Then, make sure you're using the Snake properly by tightening the locking screw once you get to the bind and spinning while pushing. You can also pull the Snake tip out to almost straight for better steering.

If still no success, then it's time to get a Plumber with a Camera Snake and not just a bigger or powered Snake. You may very well be bottoming-out at a non-directional T-fitting or 90-degree elbow. Neither of those "should" ever be present in a plumbing's drain system, but commonly are nowadays. This may therefore require Snaking backward from the street if a lower clean-out access point can't be installed to bypass the bind.