Its a question of rigidity.
The part in the chuck is nice and rigid but the part being cut off is unsupported and effectively flapping around in the wind. So when the stub gets very small its easier for it to break off at the stiffly held end than to be cut.
Narrow blades and high speeds tend to be better but our smaller manual lathes may not be able to operate in the best range. As always with carbide inserts GTR and GTL blades work best at their design speeds and feeds. Something worth checking before you buy as there are some very demanding ones about.
The traditional way to avoid a stub on a parted off part with a HSS tool is to grind the end at an angle to as to put the narrowest part of the stub up against the bit being cut off. Unfortunately this creates a side force which tends to bend the blade so you need a relatively thick blade or tool to make straight parting cuts of any depth. Thick blades are hard work for small lathes, maybe too hard.
Clive