Do I have add the font-weight in order to use it on web?

Hi,

At work we have a client that uses Typekit. They have a font with the weights 300 (slim) and 400 (regular) at the time being. The design showcases a few places where they gone need a 600/700 (semi-bold/bold) font in that particular font. Now we can do like this and it works, but it just doesn't look right, it seems as the browser is "faking it". My questions are, do they have to add the bolder version to the "web-project collection"? And, is there any truth to my theory of the browser faking it. 

Thanks

font-family: 'my-font', serifs;

font-weight: 700;