The BA website is pretty clear. You can check 2 bags per passenger in WT (plus a stroller and car seat for a child).
http://www.britishairways.com/en-gb/...age-allowances

They were trying to check 4 bags plus an additional 2 bags, hence are liable for the extra bags.
http://www.britishairways.com/en-gb/...weight-baggage

Did they also have additional hand luggage to take on board as well? The hand luggage allowance is 1 cabin bag plus 1 personal bag.
http://www.britishairways.com/en-gb/...age-allowances

On the hand baggage page it also states: If you go over your allowance of 2 bags, you may have to check in one of your extra bags, for which there may be a charge.