You can feel the cutting pressure go up. There are potentially two things going on here. First, the correct way to look at the cut is the feed per distance traveled. Not the feed per rev, since this ignores the thickness of the chip being lifted. Imagine unrolling the cut. It is like a shaper whose downfeed is held constant, but the ram speed is getting slower and slower. Eventually, the bit plows too deep and experiences excessive force. The second thing going on is that any misalignment of the tool off center becomes more serious. If the cutter is low, the tendency to roll over the top becomes worse as the stub gets thinner. If the cutter is high, the clearance angle gets worse as the diameter decreases. You can see this by just drawing a picture.
Once, an experienced machinist told me to just keep turning the cross-feed, and turn even harder if experiencing any trouble parting. This works if one is having problems with chatter, but when the diameter gets small, pushing faster and harder doesn't help. Yes, I admit it. I followed the advice blindly (you know how certain those experienced machinists can be), and I got the expected bad results. It reminds me of when I was criticizing an engineer's design. It failed in the simulator, failed on the bench, and eventually failed in the field. He grinned sheepishly and said, "that's just because you're an idiot", but at least that stupid thing never came up again.
Think through what you are doing!!!!