It's March 2018 and I'm having a similar problem with my Roomba 655. Of course I'm out of the warranty period. I bought a new $80 battery, but the problem remained--which really irritated me--so it is NOT a battery problem for me.
Things I've done to get this thing working:
--Read lots of blogs, googled and youtubed for answers, but I have found no info to help solve my problem.
--Several times I have simply opened the cover and removed and replaced the battery, which helped temporarily.
--I have held down the SPOT and DOCK buttons simultaneously for 10 seconds in an effort to reset.
--I have done the 30 second "reset the charging system" process (as mentioned in the post above by OldTurkey).
--I've talked to Roomba customer service reps 2 times--nice people--and I bought the new ROOMBA brand name battery.
So, before just sending the new battery back and chunking Mr. Roomba in the trash I decided to closely observe my obstinate Roomba 655 while in operation and noticing WHEN it stops. I believe that the battery is losing its connectivity to the battery prongs because it typically stops working and the battery light flashes amber AFTER it bumps the wall, goes over an air vent, or goes on or off a mat or area rug-- although sometimes it does just randomly stop for no apparent reason but I think this is also a loss of battery connectivity. If I simply place the Roomba back on the charging base, sometimes it will just restart normally (it's like it has to remind itself, "oh, we DO have a battery connection").
So, finally I opened up obstinate Roomba and put a buffer between the battery and the bottom cover in an effort to absorb battery movement during operation. (I used a strip of the spongy material that's normally used to keep rugs from sliding around). This has helped my problem IMMENSELY and some days my Roomba will run a complete cycle without stopping and other days it stops and I have to just press the CLEAN button or I have to return it to the base , wait about about a minute (give it time to think :) and press CLEAN . I'm still trying to figure this out and get to the point that the Roomba does not stop mid-cycle, so my next action will be to bend the silver prongs that the battery connects to.
Trying to prove I'm smarter than the robot . . . .
Today I had to restart my roomba twice, but it is now happily cleaning my floors and carpets.