Having Clause or something close

So you should be able to do this with a measure like the following

Qualified Customer = 
COUNTROWS(
    FILTER(
        VALUES('Customer Info'[Account_NBR]), -- gets a distinct list of account_nbr
        CALCULATE(                            -- forces a context transition so that Case_ID is
                                              -- filtered for just those under the current Account_nbr        
            COUNT('Customer Info'[CASE_ID])
        ) > 6
    )
)

If you had a measure that counted case_ids

Case Count = COUNT('Customer Info'[CASE_ID])

Then you could simplify this to remove the call to calculate (as measures are wrapped in an implied calculate )

Qualified Customer = 
COUNTROWS(
    FILTER(
        VALUES('Customer Info'[Account_NBR]),   -- gets a distinct list of account_nbr
        [Case Count] > 6      
    )
)