Can potato gnocchi dough be overworked or simply break?
Your second suggestion sounds like the most likely option. As long as you had a sufficient layer of flour, it worked, but once it was wetted through, they started sticking and fell apart. This is usual for many wet and tender doughs. 

Overkneading is practically unheard of in potato based doughs. In bread, you can get overkneading when you knead so much that the gluten breaks apart. It requires a well developed gluten network, which you then overstress mechanically. I have never heard of it happening during hand kneading, and even in a machine, you have to go really far to overknead. Potatoes don't contain gluten, and the wheat flour added is not sufficient for a gluten mesh to form, so there is none present to break apart. 

If it happens again, you can try kneading and forming the gnocchi in a wide shallow bowl full of flour. It will make the dough a bit harder, but you won't experience sticking.