adding to that... yes, but even if it is a ESRC quota studentship in question, the money and the place to study are technically two different things. the department might decide to give you a place but not the funding. then it would be up to you to either go somewhere else or try to find other funding. still, the place and the money is linked in that you won't get the money without the place
I think it is different with other research councils, but at least in the ESRC the open competition is also bound to departments (that's what we've been told anyway) - it is departments who put forward applications for the open competition and the number of such applications they can put forward is limited, so there's no avoiding the department.