I have a 6-string downtuned to A-E-A-D-F#-B and I used a Peavey 5150 cab (with Peavey 5150-II amp into it). At very very high volume (ie >5 on post) the speakers begin to sound a little bit flabby, but otherwise it's tight and punchy at the lows.
For what it's worth, I have resonance at full and low very high on the amp.
There could be several causes of the trouble you're having, such as (eg, and not limited to) problem with the speakers, loose fixing of the speakers to the cab, an even more bassy sound than mine (heh, unlikely :D), too much power going in (again, unlikely, this is a 320W cab!).
For what it's worth, I'm planning to get myself some Celestion G12K-100s, and possibly two other speakers (maybe Eminence wizards?) to go in the cab to replace the stock Sheffields. You might want to try checking the cones of all the speakers, also checking the fixings.
From my own experience with my 5150 cab, it's unlikely that the cab 'without a problem' can't cope with a 7-string...
Roo