Well the main question really is what you mean by you can't get a connection. Do you mean that your internet connection goes down? Or rather, do you mean that FireFox cannot load the page?
KDE's browser is called Konqueror, but you already knew that :-P.
One way you can check the above is to run pidgin In tandem with your browser. If pidgin works but the browser doesn't, well, then there's your answer. I think you see where I'm going with this.
If you're having problems with both browsers, it more than likely has to do with your DNS configuration (that /etc/resolv.conf file mentioned above). When I first installed Linux I had a similar issue.
If you're feeling adventurous, try editing your /etc/resolv.conf file to include the three DNS server IP addresses for your area, found here http://www.dnsserverlist.org
using the following syntax
# use pound symbols to comment out the current config
# then type
nameserver (IP address 1)
Even if that doesn't solve your problem, it will more than likely give you a better browsing experience than the default DNS servers provided by your ISP. Just remember to save a backup of that file and copy it to the right place on startup (usually network config programs rewrite it every time a connection is established).
Hope this helps. Happy Hunting :-P