Helping people with computers... one answer at a time.
Network failures after standby are not uncommon in older machines. Fortunately, fixing network problems after standby is not difficult - usually.
I have a Dell running Windows XP Home addition. When I put my computer in standby or sleep or if it has been inactive for several hrs , when I resume use I find I cannot connect to the internet - or the browser says "website cannot be found" like it's not connected. I use DSL and the connection works fine on my other computer with no problems. I restart the problem goes away for a short time - till it is inactive again. What gives? I have 2 hard drives on this computer. My other drive never has this problem. I have even deleted, reformatted and reinstalled windows XP and the problem is still there.
I wouldn't expect the hard drives to be a factor at all.
What you're describing isn't terribly uncommon, and points to either of a couple of places.
What you're describing is simply a lost network connection. As you say, the computer cannot connect to the internet - at all - so pages can't be retrieved, and the error you see in the browser is the result. If you have other machines on your local network, you probably can't see or connect to them either, until you reboot.
The issue here is power management. When you put a computer into standby, software is not unloaded, but left in memory while most all of the other hardware is turned off to conserve power. When you resume from standby all that hardware is turned back on, and then needs to not only be re-initialized, but it needs to be restored to the same state it was in before standby. That's when things can get confused.
As I said, there are two common causes:
BIOS - Your computer's BIOS, the software that's 'built in' to the hardware of your computer and which acts as an interface between Windows and the hardware, plays a very large role in power management. Problems with standby were very common in older laptops as the BIOS on many computers wasn't handling every scenario properly.
Your first step should be to check with the manufacturer of your computer, Dell in your case, for an updated BIOS for your machine. Installation is typically a download which you then run, or copy to a boot floppy or CD and then boot from.
Drivers - as I said, "waking up" from standby requires that each piece of hardware be re-initialized, and that's in part the job of the drivers for that hardware. You didn't mention what kind of network connection or interface you are using, but if it's drivers don't handle resume properly, the result could be a dropped network connection.
Your second step, should the first not resolve the issue, would be to locate updated drivers for your network interface. Once again this is probably something you would locate on your computer manufacturer's site - Dell, in your case.
If neither of those options work, next steps become a little more hazy. I'd certainly make sure that Windows itself was fully up to date with a visit to Windows Update.