Perris, If neighbour's annual strim is early in growth cycle (i.e. when the fronds start to unfurl) that is a good thing. If that is the case and assuming you both hit them about the same time, slow progress will be made! However, from my limited experience so far, one hit per year will not be enough.
I have yet to deal with my own bracken properly, but I am now better kitted-out with equipment and will be scheduling a couple of bracken hits a year at least.
(The herbicide option is not something I wish to consider right now - but ....... !)