Your network is wireless (802.11 wifi)?
When it says connected you mean the wifi say's its connected, right?
The key thing you need for the Internet to work is an IP address assigned to your PC. When you get this connected but can't get on the Internet check your PC to be sure it has a valid IP address. Since your routers IP is 192.168.1.1, your PC should have an IP address like 192.168.1. (any number between 2 and 254). If the IP address assigned is anything else, that means your PC just stuck in a random address on it's own. You won't get connected unless your on the same subnet as the router.
One thing that can potentially be happening, since your have multiple computers is the router's DHCP table could be screwed up because you put your computer to sleep. When you put it into sleep/hibernation the PC doesn't release it's IP address. Your router could reassign the IP address to another device on your network. Then when you wake up the PC it tries to use the IP address it had when you put it to sleep. Since the network can't have overlapping IP addresses, the router gets confused.
You can access your PC IP address through the command line prompt with the command IPconfig.
Go into the command line prompt now and type IPconfig (enter). You will see your PCs IP address information. Next type IPconfig /release. That will release your address (your connected but can't get on the Internet), then type IPconfig /renew you will get an IP address assigned again.
When you get the connected but can't get on the internet do the /release /renew trick. That should fix it. Or, don't but your PC in hibernation - shut if down or don't turn it off at all.
Follow all that?
edit ------
Is your wifi router located near a wireless telephone base station? 802.11 and wireless telephones use the same carrier frequency and they can step all over each other. If you have a wireless base station move it far away from your computing equipment.
Can you connect your PC with a wired Ethernet connection on the router? The reason to try this is to see if the problem follows the router or the wireless.
Could be your router is crapping out.