Whilst Felicity is almost correct in saying you can't have more than one account, this is 90% true, however it can be possible when you first start applying for a student account to apply for multiple (usually in your first year)
A TV programme conducted an experiment in the last year which highlighted that this was possible with some banks offering large overdrafts providing a huge credit facility for most students.
It is possible to open more (non student) accounts, but if you want to have an overdraft or other credit, you would need to provide an income, and this is where the student accounts differ, this would need to be a month regular income, whereas it wouldn't for the student account.
You could apply to have the second account as you have suggested, but your current bank may be able to see this on your credit record when they conduct a credit search to open the account, and may decline because of this.
HTH
Sparky
PS I'm guessing you are a RATM fan