I think this works:
=SUM(COUNTIFS(Accounts!B:B,A4,Accounts!F:F,"Medcare Company",Accounts!I:I,"Yes"),COUNTIFS(Accounts!B:B,A4,Accounts!F:F,"Inv Lab",Accounts!$I:$I,"Yes"),COUNTIFS(Accounts!B:B,A4,Accounts!F:F,"Physicians",Accounts!I:I,"Yes"))+COUNTIF(Accounts!I:I,"Yes")

I get 33 but I counted up the extended hours and it looks like just 28 of them plus the 5 specific carriers wihch gives me only 33.

I tried, Hope this helps.