If I remember it correctly then this is not possible at the moment.
I wrote an extension to the BI functions exactly with this purpose.
Unfortunately was this with CMK version 1.4 and I had no time to test/port it to 1.5 or 1.6.
The output looked like the BI status plus the problems inside the BI. Only if a part of the BI gets warning or critical state.
The image is a host aggregation with a warning filesystem. If you now use this aggregation as a real service you will also get the cause of the problem with your message.