I have a current and deposit account at HSBC. A few years ago my relationship manager there set up an arrangement whereby I maintain a set balance (say £1,000) in my current account. This coupled with a daytime facility to go overdrawn upto a set limit means I am able to push all payments through my current account without ever worrying about going overdrawn. It clears before midnight every day by transferring money from (or occasionally to e.g. just after I am paid) my deposit account.
Provided the deposit account is maintained with a reasonable buffer the system works perfectly. Come to think of it, given the negligible interest rate I get on my deposit account, the system could work just as well with 2 current accounts.