This happens to me from time-to-time (actually several times a week I'm unhappy to admit and I too have a Dell Inspiron - E1705), but the solution (at least for me) is relatively simple if annoying. It's rebooting the MODEM - not the router or the computer. If you're setup like me, you have a modem from the ISP connected to your router and your computers and such all connected to the router. Rebooting the computers and/or the router may not be working because they don't actually reset the connection with the ISP (unless the router is one that is directly connected - in which case ignore everything in this post about a modem and focus on the router instead).
Rebooting the modem will reset the connection with the ISP and generally works to get me back up and running (and rebooting here simply means turning off the modem power switch or unplugging it or disconnecting the cable to the wall jack, waiting 10-15 seconds (DO wait - don't rush it as it needs time to reset), and then restarting it again). If it is in another room or on another floor (as in my case), make sure all the lights come back on properly before you leave (mine takes about 30 seconds from turning on the power to being fully functional and connected so it's not that long a wait).
Depending on how you are setup, you will automatically be reconnected to the internet when the modem is functioning again or you may need to re-login to the broadband connection. In my case, it occurs automatically (but with the modem I had 3 months ago, I needed to re-login every time this happened - so at least that's easier now).
I don't guarantee this will work (it does for me most of the time, but sometimes I need to do it a couple of times before being successful and sometimes even that doesn't work - in which case it's almost always a problem with the line that I can do nothing about but wait for it to be resolved. If it goes on more than a hour or two, I usually call the ISP to report it (so they'll have someone test the line and remotely test the modem for problems and if they find none - and they usually don't - they create a trouble ticket and I merely need to wait for it to resolve itself).
They almost always try to blame my computer and want to delete my history and temp files and cookies and all sorts of other things until I point out that the problem is occurring on all my computers and started at the same time and so it isn't a computer-specific issue. That usually simplifies the process and since it seems to be the case with you as well, be sure to mention this early if you need to call them so they don't waste time trying to fix a computer problem when it obviously isn't if both are affected at the same time (and recover at the same time).
In short, try rebooting the modem a few times to see if that resolves the problem and, if not, contact the ISP for assistance. With it happening to both computers at the same time, it's either the router, the modem, or the ISP/line/network servers and not the computers.
I hope this helps.
P.S. If you're using DSL and have filters attached to all your phones and are using a splitter on the jack to which the modem is connected so you can use it for both the internet connection and as a regular telephone line, the splitter can sometimes go bad (not very often, but it can happen). They are very cheap and I would replace it at your earliest convenience to see if that helps - it certainly won't hurt.
P.P.S. If it happens more often than you can tolerate, consider changing your ISP and/or replacing the modem and/or router. I honestly don't think you can do anything on the computers for this problem as I don't believe they are causing it or can be configured to avoid it.