Questions about background check

Got questions to ask friends here：

I already signed on the offer letter and faxed it back, they say they have received it already. My questions are:

They are still in the background check (not reference check) stage, I think that could be the only possible factor that I should worry about. I worked for 2 companies and have 2 ex-bosses, one is not working in that company anymore, the other is usually super busy and responds very slow (he could possibly set this background check as very low priority work). Will these affect my background check? What if the background check is delayed or not finished before my planned start date?

Please enlighten me and thanks in advance.