Try going to a public wifi hotspot such as Mcdonalds or similar and see if the problem still exists. If it doesn't then you know it's something inside your home interfering. I have a netgear N900 router with 3 PC's, 3 phones, and a TV that run through it. Nothing stops, skips nor stutters during streaming now but the S3 I had (traded it in for the One) would do just as the OP described but only in my house. If I took it to Mcdonalds and used it there...I could use it all day long without one hiccup, but as soon as I got home trouble would start.
After intense trial and errors, it ended up being my cat 6 cable running from my modem to router. You would think my other peripherals would have cut out too, such as my TV, but they never did. My guess is that the line of sight between TV and router is unblocked with nothing to interfere. But it still didn't explain why my other phones never cut out. My biggest guess is that packets of information get lost at times during transmission between phone and router, especially if the signal is degraded between modem and router. Once I replaced the cat 6 cable, my S3 never had a problem.