The only thing I could think of was the powerline adaptor I use becasue after I turned it off I lost connection. Is it possible to connect through one of these to someone elses connection.
If they use similar powerline adaptors to yours then it's possible, yes, but not common- usually neighbours are on different phases and there's enough cable and sundry kit between the wiring systems to prevent a decent connection.
But it can happen... and if you can "piggy-back" onto their LAN, they can in all likelihood piggy-back on to yours!
Most powerline adaptors have some sort of security systems that can be enabled to prevent this happening, it might be an idea to investigate yours.