As above, the problem is not opening a new account with your existing bank, but opening an account with a new bank. In theory, this shouldn't be a problem, but in reality many will turn you down. The only answer is to use Moneyfacts
to find the best rates, and then ask each provider individually.
On a separate note, don't invest more than £35,000 with any one bank (that's how much the government will cover you for should the bank collapse), and be very wary of investing with any foreign banks
, especially the Icelandic ones.