My gut feeling is that this is a problem with either your ISP or your router.
Many consumer routers do not support more than a specific number of connections. Are you using a Torrent client (BitTorrent, Azerues, KTorrent, etc.)? Are you running a P2P file sharing program (LimeWire, Ares, Morpheus, etc.)?
Also, many ISPs will block this type of traffic. While they usually block the port used by said application type, the could potentially just hang your line for a few minutes and say that you are using too much of their bandwidth.
Either will lock up many consumer grade routers. However, all is not lost. There is quite a large community of individuals who seek to surmount this type of problem.
For example, if you have a WRT-54G series router, you can update it's firmware to allow it to support more connections (though your family will notice this change as their router will not be configured the same way again.) There are other consumer devices like this, you might have one and just not know that a huge number of people have decided to discombobulate it to the extreme and make a toaster into an airplane, or a jeep or maybe even a Yahoo server ;)
If I were you, I would remove the router from the equation. Just plug your computer directly into the modem used by your ISP. If it still hangs every 5 minutes or so, then you know that the problem will necessitate a call to your ISP.
If that does not solve your problem, it is with your router and your ISP will be of no help at all. If this is the case, then Google your router's model number (I.e. WRT-54G) and see if there is a community that modifies it.
WARNING: YOU CAN PERMANENTLY DESTROY YOUR ROUTER BY FLASHING IT
If you do find that your router has some type of cult following then there is probably a community that would like to increase its functionality. If you decide to follow this course of action, I would talk about it at dinner one day as you can PERMANENTLY DISABLE YOUR ROUTER if you do not follow the procedures precisely.
Hope that helps.