I usually run six tabs at once, and never heard of any problems from doing this. I have family in the IT business, and they do it as well.
I use Firefox, and can upload on two sites and read on another, while having music playing from Pandora. If you are having issues, it is likely a bandwidth issue, and not the tabs. I know in the days of "dial-up" running more than two was an issue, but with broadband, that is no longer the issue.