This might only be a contributing factor to the problem, but the "surface feet per minute" (SFM) cutting speed of your parting operation will vary with the ever decreasing radius, as the cutting tool moves in towards the centre of the bar.
What is a sensible calculated cutting speed for your material at the outside of the bar may be completely wrong by the time you get near the middle of the bar.
If the SFM for the tool and material goes too low then it will start creating the wrong shape chips and it could clog the slot, overheat the tool, or start rubbing rather than cutting. If this causes metal to start adhering to the tip of the cutter, then as it is in a narrow slot it can cause it to jam and snap.
Modern CNC machines are smart enough to continuously adjust the spindle speed during a parting operation to keep the SFM a constant.
On a manual lathe you might want to consider emulating this by pausing the parting cut half-way through, withdraw the tool, then increasing the RPM to finish the parting-off operation.
The other side of this coin is feed rate. This also affects proper chip production.
Check the shape and colour of the chips, and see how they change as you do your cut.
That should tell you what's going on.