I've had to fix quite a few hard wood floors and varous apartments I've managed.
The other are right in that the finish is not sticking in though sections.
I've found in general the sanding and polyurethane coats are pretty forgiving on sanding inbetween coats. IE not sanding between coats will not casue the finsh to come up like that.
But a big problme I found with old floors is carnubo wax.
Back in the old days it seemed they would wax the hell out of these floors, after 30,40-50 years of this the wax seems like part of the wood, even after light sanding.
any finish will not stick to carnubo wax. I would hazzard to say that parts were the finsh is pealing are areas that had very thick layers of wax and the light sanding didn't take it off, but remember 40 year old wax looks like wood so it's easy to miss.
I would suggest re sanding thoughs areas again and try touching up the finish rather then redoing the whole floor