We had lovely neighbours and we were more friendly that 'just nice people who live next door' we also held back on getting too close... I looked after my neighbour when he was bedridden and terminally ill when his children couldn't be there and in between McMillan visits, we adopted his cat, he was a great neighbour... close enough to bring joy and happiness to each side, not so close that he just wandered in the house and made himself at home like my close mates would.
Our new neighbour is also bloody lovely, we share many interests, always say more than hello, could get closer but I do think that TIME will see to that, I hold back a little because if you get it wrong a BAD neighbour or a fall out could become unbearable.
I have no doubt that, had he lived, our last neighbour would have become close enough to always get an invite to our family events and parties... I think unspoken ground rules are important, he knew never to knock if the kitchen light was off.
On the flip side, my friend has a neighbour with children of similar age to hers, in my opinion they got too friendly too quick, and now the neighbour invites herself around when my friend is entertaining/chilling with other friends, gets offended if she's not invited, won't take hints, and though she's lovely, many of the circle have backed off visiting my friend because she is always always there.... she's a 'good neighbour' but she's become a bit too close for comfort for my mate.