A few things to think about that may help. After 16 years, you should expect fruit.
Assuming that it has a good structure of main branches, you will need at least some shoots coming from the main branches, to form fruiting spurs. So some need to be left.
How are you cutting them? If you cut them right off, there will be no chance for fruiting spurs to develop, try cutting them down to 3 buds, and do this again if they regrow.
When are you cutting? If you cut in the winter, you will encourage more vegetative growth, and you want fruiting growth. Cut around late August, and you won't get so much unwanted growth. 360 is an awful lot to cut out of a tree.
Do you get any flowers at all on it? If you don't get many, it's almost certain that you are cutting off the potential fruiting points as ceres says. If you do get flowers, is it flowering at the same time as the other tree? If it isn't, then pollination would be difficult.
Are the flowers particularly early, if so, they may be being affected by frost. Assuming your neighbour's tree fruits well, it's possible that yours needs more shelter if frost could be the problem.
You may find it helpful to leave these shoots alone this year. See what happens next year, see how many flowers form, and where they are. If few flowers show, then leave it another year and more flowers should form the next year, and you will be able to see clearly the fruiting spurs that have developed. At theat stage you could loosely tie string around shoots that you feel are unwanted, then you could cut those around end of August.
