Last week I opened a student account with Santander, and got the Railcard sent home. But before even using my card i found out about HSBC's deal (1year prime + £60 voucher) so I went to a branch of Santander and had my account closed. Since i hadn't used the card (just the online services, no money in/out) it took about 10 minutes to close it down. Then I applied for HSBC and still waiting for a reply.
So yes you could do it, just close one of them down. If you want to use HSBC then apply for Santander first, and if not the other way around.