I had this problem on SL and on Lion as well, and it occures for WiFi and Ethernet connections.
In fact the internet connection was not broken as I could ping the destination usin the IP (not DNS name).
In this case the problem was related to mDNSResponder task which handles all dns queries.
It was the most frequent problem in my day to day work with SL and now Lion. It was never fixed by any patch from Apple. I tried almost all voodoo including fresh install, all possible config changes - no solution.
The only version with no problems related to network was 10.5.8. All the other versions had issues of different kind (WiFi instability (lost signal), DNS problems (for wireless and wired connection) sometimes associated with machine sleep and wake up and sometimes with no reason.
QUESTION: Is your problem independent of the access point / router you connect to.
I must say networking was (and still is) the weakest part, besides the other OS X weaknesses I experienced during 4 years of OS X usage.
No problems in the same network with Linux or Windows beeing ran natively on my iMac 🙂 So hardware is OK.
Software is of poor quality. Not much more to say in this matter.
I found a very interesting article dated 1996 about programming which I think is still valid.
Link to the article: http://www.fastcompany.com/magazine/06/writestuff.html
Below the fragment (with some connection to Apple)
Software is everything. It also s***s.
"It's like pre-Sumerian civilization," says Brad Cox, who wrote the software for Steve Jobs NeXT computer and is a professor at George Mason University. "The way we build software is in the hunter-gatherer stage."