I know that VISA and MC have no objection, though most banks that I know about only offer one or the other as a
debit card. But it is common, especially in the UK, for them to offer both as a credit card, many people have a matching VISA and MC card from the same bank. Maybe one day they'll be allowed to put both logos on the same card?

(It
has been suggested in the UK)
I think that VISA and MC used to have a restriction clause (in the UK) - so that one bank couldn't work with both, but that was overturned in the courts as an unfair restriction of trade.