The rejections (which were "instantaneous" online ones) included a message saying they would send us the reason for rejection. I don't remember the timeframe, I think within 30 days. I will probably call them if the reason seems off base.
1moremom, my understanding is that all Capitol One credit cards have no foreign exchange fees. They even waive the Visa/Mastercard portion of the fee (2% of the usual 3% you get charged); well, waive isn't the right phrase, they actually eat it.