Well opening a new one will bring a credit check. So if you are in your Overdraft they may just reject it. Usually a term of condition for student accounts is you will only have one open (regardless of bank) and use it as the main account.
Obviously not doing that could cause your old account to get closed too.
But you can always try, and can always see if the current bank will raise your OD.
Can you potentially not get aid from family ?