When you say the internet goes out all of the time, do you mean that you lose connection to the linksys, or that the linksys loses connection to the internet?
If it's the linksys losing connection to the internet, there are a few things that you can do, a few of them involve social engineering. Such as, asking them if they too, lose connection to the internet daily / hourly / whatever. Implant the seed of doubt in their ISP, have them investigate it. Old people always think that they're getting screwed, play on that.
If you're losing your wireless connection, then something is causing it. Do you / they, have / use a microwave often? Microwaves circulate through frequencies and will completely fuck a wifi signal. What is separating your two homes? Sheetrock? Stucco? Stucco uses chicken wire and is notoriously shitty for external wifi connections, same goes for brick and mortar. Do you or the neighbors have 2.4GHZ or the 5.whateverghz phones? The 5.whatever do actually use a whole bunch of bands in the 2.4GHZ range that can effect a wifi signal. Also, what other wifi points are around their AP? If they're on the same channel as the other neighbor across the street or next door, there will certainly be some interference, change the wifi channel on their unpassword protected linksys. Usually a two channel spread is ideal, but I like to jump to the other end of the spectrum.
The best thing for your connection is to move as close as you can to their AP, and stay there, if it's still fishy, change channels, or boost the signal strength (you can do that with some linksys's, netgears, and 2wire [it's for DSL, but it's helpful to know] modems). If not, get/make an antenna/cantenna.
