I am also having this problem with my new 2013 Macbook Air 11-inch (1.3Ghz, 4GB), purchased only 3 days ago. My wifi connection connects okay at first, but after anything from 1 minute to an hour, my internet disconnects, even though it shows that the wifi signal is at full strength. Turning wifi off and on from the top bar gets it working again, but before long it will lose the connection as above.
My router is a Buffalo Airstation WHR-G125 running DD-WRT. All of my other devices are connected fine (Macbook Pro 2008, iPhone, iPad, Roku). Haven't tried the connection anywhere but home yet, will try to report back after I have. Frankly though, even if it works at other locations, there's no saying when it may happen across another router that it doesn't 'like'. It's simply not good enough.
Now I have to look forward to either forking out for an Apple router, which according to posters above seems to have solved the problem at home, or going on a happy merry go-round with infuriating Apple telephone tech support and Geniuses, all of whom are probably going to tell me that the problem is with my set up (again going by the experiences of posters above).
If I could, I would just take it back for a refund, but unfortunately it was purchased in Hong Kong (the 2013 Macbook Air isn't scheduled for release until December in Mainland China!!!), and I would have to go to HK to get it.
This is the second Macbook Air I've bought, the first one needed replacing twice and parts replacing another two times thanks to various hardware problems. You'd think I'd have learned by now...