That is very odd.
As far as the machine is concerned, you aren't connected to the net - it is aware that it has a network card, but sees nothing in there in terms of media.
What is very weird is that sounds like a hardware issue, like a faulty cable or jack on the card - but were that the case, it wouldn't matter what browser you used.
My first guess is that it would be a virus or trojan, but it wouldn't make sense for those to sever your net connection because then they could not spread any further.
Were it just a hardware issue, it also wouldn't likely come and go over time - it would likely be due to jiggling the cable or something.
That said, if there is a chip on the card that is overheating (or some component nearby giving off heat), then it would be a time based thing - basically until enough heat built up until it gave out due to stress.
But again, this would happen regardless of the browser used.
How is it that you are on the net? You have to restart the machine?
That too is bizarre and would point to a software issue such as a virus...