I just called the Bed Bath & Beyond Rebate Center and they told me there is no limit t the number of rebates you can submit. That being said, everyone is being told different thing by different agents so who knows if there really is a limit and what the number is. We will find out if/when peoples rebates start getting rejected.
That being said, to respond to the point about doing split payments with multiple happy gift cards, if you start doing multiple split payments you have a much better chance of them asking to see the cards you are using or them taking notice of the cards you are using and getting shot down. People have been shot down for trying to use one happy card, if you try to use 3 or 4 on one transaction... who knows.