It's not so simple as it sounds, especially with new versions of firefox/chrome released all the time, major changes to IE between versions, and ohh wait... there is also safari, opera, etc. etc.
It also a lot of the times depend on which extensions you have installed, what cookies are stored on your pc and many more factors, so even if a someone will test the website functionality for a week he will not be able to detect an issue an end user will see after doing some sequence of actions etc.